Because of an edit made in your post, it is important to note that money itself is not being called the root of all evils (nor all sorts of evils) in this passage, it is the love of money that is problematic, as the edit to the question has clarified.
With that said, 1 Timothy 6:10 is a difficult text to translate. A literal translation of the text would read like so:
For a love of money is a root of all evils; some [people] in
reaching for it have strayed from the faith and pierced themselves
with many sorrows.
My initial thoughts concerning this translation is that using a literal translation makes this verse less 'quote-worthy'. People like to quote individual verses in the Bible without their surrounding context. If you don't use the definite article and helping words, this verse doesn't make much sense standing alone. This makes it less 'quote-worthy' unless one also reads the surrounding context.
I don't believe making a verse into something easier to quote is a good motivation for adding helping words to clarify it, but it also isn't really changing the meaning in context, it is merely adding terms to clarify it.
Translating the indefinite noun ῥίζα
Despite this literal reading, translators agree virtually unanimously on translating 'root' as a definite noun in this passage, and as you have observed, they often disagree on whether to translate πάντων τῶν κακῶν as 'of all kinds/sorts of evil/s' or literally as 'of all evils' (based on a survey of major English translations). However, using the definite article often forces these same translations to add 'kinds/sorts' to avoid the text making the (obviously) false assertion that the love of money is the root of all evils, which I believe is the motivation behind inserting these helping words.
Daniel Wallace (who was one of the principal translators on the NET Translation Committee) points out in Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament that there are six (6) possible readings of this verse (p. 265):
- “the love of money is a root of all evils”
- “the love of money is the root of all evils”
- “the love of money motivates all evils”
- “the love of money is a root of all kinds of evils”
- “the love of money is the root of all kinds of evils”
- “the love of money motivates all kinds of evils”
The reason for these six possibilities is that first, it is difficult
to tell whether ῥίζα is indefinite (options 1 & 4), definite (2 & 5),
or qualitative (3 & 6), and secondly, πάντων may mean “all without
exclusion” (1, 2, & 3) or “all without distinction” (4, 5, & 6).
Logically, it would be difficult to say that ῥίζα is definite, for
then the text would be saying either (1) the only root of all evils is
the love of money or that (2) the greatest root (par excellence) of
all evils is the love of money. These are the options if πάντων means
“all without exclusion.” However, the definite idea would fit if
πάντων means “all without distinction.”
Grammatically, it would be difficult to take ῥίζα as indefinite, since
this is the least attested meaning for the anarthrous pre-verbal
[predicate nominative] in the NT. However, grammatically the most
probable option is to see ῥίζα as qualitative. The idea would be
either that all evils can be motivated or initiated by the love for
money or that all kinds of evils can be motivated by the love for
money. The qualitative idea makes no comment about anything else that
might motivate or produce evil. It simply states that loving money
does motivate/produce all (kinds of) evils (p. 265).
Despite Wallace's conclusion based on the analysis of this particular text, it is also notable that the NET translators point out that "there is no parallel for taking a construction like this to mean 'all kinds of' or 'every kind of.' The normal sense is 'all evils.'" The next section of this post will thus analyze the occurrences of πάντων τῶν κακῶν elsewhere.
Also notable is that by his own admission, Wallace's analysis only consisted of similar constructions in the New Testament corpus. This can often produce a short-sighted view of the linguistic use. A fuller study should compare similar grammatical constructions in all contemporary and other early Greek literature. Unfortunately, I do not have the time nor inclination to conduct such a study for an answer on BH.SE (such an answer would be for publication in a peer-reviewed journal or book). Even so, a cursory search for πάντων τῶν κακῶν was conducted of classical Greek literature (still not the ideal corpus in all cases, but it can still shed light on this).
Delimitation: I am only searching for the phrase πάντων τῶν κακῶν in the Perseus Digital Library. I am not searching for all anarthrous pre-verbal predicate nominatives to see if they are best understood as definite or not, nor am I searching for other key terms and phrases within the verse (doing so should be done for a complete analysis of this issue, but is the stuff of dissertations and academic publications, not answers on SE).
Occurrences of πάντων τῶν κακῶν in other Greek texts
1 Timothy 6:10 is the only occurrence of this phrase in the New Testament corpus, however, the phrase πάντων τῶν κακῶν occurs in six (6) verses in the Septuagint (LXX):
All of the occurrences of πάντων τῶν κακῶν in the Septuagint appear to best be translated as 'all the evils'. Many instances occur with a genitive pronoun (e.g. 'all their evils'), but even so, none of these instances seem to be best understood as 'all sorts/kinds of evils'.
Next I searched the Perseus Digital Library for instances of this phrase in other early Greek literature and found twenty-four (24) occurrences (I did this search with software, but you can do the same search online for free, this will link directly to the search results; note that the text of interest was one of these occurrences, i.e. 1 Timothy 6:10). I only took a cursory glance at the results due to time constraints (although I did personally translate several of them, but my analysis of the first ten results was admittedly much more thorough than that of the last fourteen). In most of these cases, the more literal translation also seems preferred (i.e. not adding 'kinds/sorts of').2 The few where adding 'sorts/kinds of' may be preferable for translation (the English translation on the Perseus Digital Library opted for the literal translation in these as well—this is more or less a concession on my part from my analysis of the Greek text) are listed below:
- Isocrates, Against Callimachus 18:11 — Isocrates refers to Xenotimus as the 'author of all evils', which is a problematic assertion without qualifying it with 'sorts/kinds of' as in 1 Timothy 6:10. Even so, it can be translated literally as hyperbole, but this at least argues for translating πάντων as 'all without distinction.'
- Lysias, Against Simon 3:20 — Lysias calls Simon the author 'of all the evils', but in context these 'evils' seem to have an immediate referent, allowing for a literal translation (which may also be the case in the letter to Timothy).
I could spend a lot more time on this, and one probably should to better grasp this issue. But based on my cursory analysis, I'll conclude with some of my thoughts (I here assume knowledge of the textual context, as I did not specifically address it in my answer):
- The point of this passage in context seems to be that loving money motivates or produces evils.
- Whether the helping words 'kinds/sorts of' should be added depends on how you understand the context. A literal reading could be supported without these helping words if it were an intentional hyperbole or if you understand 'all the evils' to refer to the 'harmful desires' that cause people ruin and destruction (v. 9), as well as other earlier listed vices. However, not using the definite article and helping words makes the text less 'quote-worthy' and thus more dependent on its surrounding context (which isn't a bad thing in my opinion).
- Wallace's analysis of the New Testament corpus seems to jive well with my cursory analysis of other Greek corpora. That is, the definite idea fits since πάντων can mean 'all without distinction' in Greek literature.
- There is some ambiguity between saying that 'all evils can be motivated or initiated by the love for money' or that 'all kinds of evils can be motivated by the love for money' (i.e. the translation difference does not result in a huge change in meaning either way).
- I believe that the context should be the final guide, which leads me to support the following translations, the first of which I believe a footnote should help explain, as demonstrated below:
For the love of money is the root of all evilsa. Some
people in reaching for it have strayed from the faith and pierced
themselves with many sorrows.
a 'all evils' in this passage is either an example of
hyperbole or it refers to those 'evils' listed in vv. 4-9.
For the love of money motivates/produces all kinds of evil(s).
Some people in reaching for it have strayed from the faith and pierced
themselves with many sorrows.
I am aware of some non sequitur logic in my final analysis, however due to time constraints I decided I needed to end this answer. I hope the work that has been done is a good start for your continued study of this passage.
1 Alternate reading labeled 10:3f in Alfred Rahlfs and Robert Hanhart, eds., Septuaginta: SESB Edition (Stuttgart: Deutsche Bibelgesellschaft, 2006), Es 10:3f.
2 I originally thought that Demosthenes' For the Megalopolitans 16:24 (c. 352 B.C.E.) contained a similar construction as that in the first letter to Timothy (c. 62-76 C.E. at the earliest), or properly said, I thought the letter to Timothy contained a similar phrase as that found in Demosthenes' political oration. However, only the English translation was similar, the Greek text itself was dissimilar except in the use of the phrase πάντων τῶν κακῶν. The specific Greek text of interest is "...καὶ ταύτην ἀρχὴν οὖσαν πάντων τῶν κακῶν, τὸ μὴ 'θέλειν τὰ δίκαια πράττειν ἁπλῶς", where the English translation used by the Perseus Digital Library has "...the root in fact of all evil—is unwillingness to act justly under all circumstances." However, a more literal translation would read "...and originates all evils, is the unwillingness to observe justice uniformly." Retrieved from http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:greekLit:tlg0014.tlg016.perseus-grc1:24.