No. I'd say, grammar-wise it would be OK, to relate 'faulty' exclusively to 'ministry'. But this is logically not convincing because of the context and sequence. Concerning the language on a basic semantic level. The language on the interpretative level however changes this with high probability into a non-issue.
The apparent ambiguity causing words
the ––– first [one]
Art-NFS ––– Adj-NFS
Referring either to,
the first entity:
διαφορωτέρας ––– λειτουργίας,
more excellent ––– a ministry/service
Adj-GFS-C ––– N-GFS
or th second entity:
κρείττονός ––– διαθήκης
of a better ––– covenant
Adj-GFS-C ––– N-GFS
Both of the same type, only closeness in word order giving a hint.
To follow that in English with YLT
(6) and now he hath obtained a more excellent service, how much also of a better covenant is he mediator, which on better promises hath been sanctioned, (7) for if that first were faultless, a place would not have been sought for a second.
In here, we see the same ambiguity appearing. But is it one?
The original Greek likely didn't use any discernible punctuation. Therefore, conventions, context and rhythm can be used to glimpse into the structure of meaning.
– νυν[ὶ] δὲ διαφορωτέρας τέτυχεν λειτουργίας,
– ὅσῳ καὶ κρείττονός ἐστιν διαθήκης μεσίτης,
– ἥτις ἐπὶ κρείττοσιν ἐπαγγελίαις νενομοθέτηται.
– Εἰ γὰρ ἡ πρώτη ἐκείνη ἦν ἄμεμπτος,
– οὐκ ἂν δευτέρας ἐζητεῖτο τόπος.
It is really a run-on sentence structure where "the more excellent service" is seemingly subject to – or serving, I might add – "the better covenant".
– and now he hath obtained a more excellent service,
– how much also of a better covenant is he mediator,
–– (which on better promises hath been sanctioned, )
––– for if that first were faultless,
––– a place would not have been sought for a second.
From the context of the preceding verses we already have to infer that "he" is Jesus, and his qualities were contrasted with that of the previous priests and Moses. The "better promises" fragment is just a qualifier for "the covenant" and the next lines equally expand on that.
What seems to follow is:
Now, however, he has received a more excellent service to the extent that he is also a better mediator of a better covenant (which has been established by law on the basis of better promises.) For if the first were blameless, not for a second would a place be sought.
But even more interesting it gets once you read the composition more rhetorically and give a little poetic licence to a hendiadyoin: 'better service' and 'better covenant' form a pair of words that are really almost identical twins. In this way, there is really not much sense to ask whether it is either one or the other.
Sansone [and his] conclusion on what he views as hendiadys in Greek is that
“it coordinates two elements, either of which could be logically and grammatically subordinated to the other” (italics Sansone), and that the main characteristic of hendiadys in Greek is its “reciprocal quality,” which makes it possible for the writer to “convey simultaneously the immediacy of co- ordination and the logical precision of subordination.”
Rosmari Lillas: "Hendiadys in the Hebrew Bible", Dissertation in Religious Studies at the University of Gothenburg, 2012.
That may be my personal view here, but it seems as if adding the word covenant into 8:7 changes little of the superficial meaning, but this insertion takes away what it wants to give.
Even discounting the theory of exact classification as a hendiadyoin, the two concepts of 'old covenenant'/'old earthly priesthood' and 'new covenant'/'new heavenly priesthood' or both correlated repeatedly. Both are inherently linked and thus juxtaposed, forming one dichotomy with two elements each which are not really conceivable as distinct.
(Cf Brian C. Small: "The Characterization of Jesus in the Book of Hebrews", Biblical Interpretation, Vol 128, Brill: Leiden, Boston, 2014. (p 194f, 250f)
This introductory pericope closes, as did the exordium (I :4), with a comparison. The degree of the superiority of the liturgy Christ performs is correlated with the superiority of the "covenant" he inaugurates. This particular correlation is at first sight surprising. Although the theme of the covenant had been briefly mentioned in the previous chapter (7:22), it has not been developed and its relationship to the institutions ofcult and priesthood is unclear.
Why that correlation is made in Hebrews will become clear as the covenant theme develops.
The first clause of the comparison summarizes the perspective of the preceding five verses. Christ as heavenly High Priest has obtained a "superior" ministry. The adjective is the same as that used of Christ's name at I:4. The nature of Christ's "ministry" is not discussed. That he exercised a heavenly ministry is a formal assertion based upon the deductive logic of vs 3. The introduction of the theme of the covenant is one step in the process of giving that assertion its content.
The second half of the comparative sentence looks forward rather than back, twice using the characteristic "greater."62 Christ is the "mediator" of a superior covenant. In Hellenistic legal terminology a mediator was any sort of arbiter or intermediary. In Judaism various mediators were envisioned including intercessor angels and the spirit. The primary mediator was, of course, Moses in his role as agent of the Sinai covenant. That Christ fulfilled the mediatorial role in a special way was probably a traditional Christian view.
The covenant of which Christ is the mediator is "greater" because it is based or "enacted" in contrast to the Law, on greater "promises". The juxtaposition of Law and promise recalls Gal 3:21, where Paul denies any strict opposition between the two. Our author, because of his construal of the Law as related to cult and his denigration of that cult (7: 11, 19), is more radical.
The introduction recalls in both form and content the earlier critical remarks on the Law. If the first were "blameless" another would not be introduced. The expression used for that introduction, "a place would not be sought" involves a common Hellenistic metaphorical use of"place." That the first covenant was not beyond reproach is, like the Law's ineffectiveness (7:11), inferred from a scriptural promise.
–– Harold W. Attridge: "The Epistle to the Hebrews", Hermeneia – A Critical and Historical Commentary on the Bible, Fortress Press: Philadelphia, 1989. (p215–229)
For the purpose of dissecting the grammar the above commentary most strikingly divides Hebrews into distinct blocks between 8,1–6 and 8,7–13. A logical and narrative break in the flow of argument like that would emphasise even more that 'the first' would not be referring to just one single word alone.