Short Answer: After weighing all of the evidence (both internal and external), it would seem that Mark 16:9-20 was indeed originally part of Mark's Gospel.
The ending of Mark's Gospel is one of the major textual problems in the New Testament.1
As noted by the OP, the "problem" is whether the end of Mark (16:9-20) was originally part of Mark's Gospel. Essentially the matter comes down to three factors.
1. External evidence
Dr. Constable summarizes the external evidence concerning verses 9-20:
The two oldest Greek uncial manuscripts of the New Testament . . . plus many other old manuscripts, do not contain them. Moreover, the writings of some church fathers reflect no knowledge of these verses. On the other hand, verses 9-20 do appear in the majority of the old manuscripts, and other church fathers do refer to them.1
DETAILS AGAINST: absent from א, B, itk, Sinaitic Syriac, ~100 Armenian mss, the two oldest Georgian mss. Some mss that do include it mark it with asterisks or obeli. Clement of Alexandria and Origen show no knowledge of the existence of these verses.3
DETAILS FOR: too many to list!!4... 99% of Greek manuscripts contain it.5 present in the vast number of witnesses, including A C D K W X Δ Θ Π Ψ 099 0112 f 13 28 33 al. Numerous patristic witness beginning with Irenaeus and the Diatessaron.3 Found in many early versions: Vulgate, Syriac, Peshitta, Coptic, Sahidic, Bohairic, Fayyumic, Gothic, Armenian, Georgian, and Slavonic.6
Despite the absence of these verses in some old manuscripts, and the silence on these verses in the teachings of some church fathers (which could be said for any verse in the New Testament), the fact remains that most old manuscripts do contain these verses, and many church fathers do refer to them. Dr. Constable concludes well that:
[these verses] appear to have been regarded as inspired and therefore authoritative early in the history of the church.1
Thus, while external evidence in this case is far from conclusive, the evidence does weigh in favor of these verses originally being part of Mark's Gospel.
2. Internal evidence: structural coherence
Dr. Constable mentions an important structural consideration in this debate:
Throughout this Gospel, we have noted many unique features that appeal to disciples to serve God by bearing bold witness to Jesus, even in spite of persecution and suffering.1
This makes sense in light of Mark's target audience. Mark wrote his Gospel to disciples in Rome who faced a serious decision:
They faced the choice of whether to take a public stand as Christians—and suffer the loss of real estate, personal property, employment, and even their lives—or to lay low. They were required by law to offer a pinch of incense in worship of "divine" Caesar, as Roman citizens. Doing so compromised their exclusive commitment to Jesus as Lord. To fail to worship Caesar cost them dearly.1
Mark's Gospel served as an exhortation to these Roman disciples to lay down their lives and preach the gospel with boldness. In light of this, it is hard to imagine Mark ending his Gospel "with an example of disciples too fearful and amazed to bear witness to the resurrected Jesus."1 Yet this is exactly what we would have if Mark 16:8 were the end of the Gospel. Thus, structural coherence also weighs in favor of Mark 16:9-20 originally being part of Mark's Gospel.
3. Internal evidence: stylistic clues
Some interpreters believe the vocabulary, style, and content of these verses argue against Mark's authorship of them.1 (E.g. Walter Wessel and Robert G. Bratcher and Eugene A. Nida.)
Bruce Metzger elaborates on the two main considerations:
(a) The vocabulary and style of verses 9-20 are non-Markan. (e.g. απιστεω, βλαπτω, βεβαιοω, επακολουθεω, θεαομαι, μετα ταυτα, πορευομαι, συνεργεω, υστερον are found nowhere else in Mark; and θανασιμον and τοις μετ αυτου γενομενοις, as designations of the disciples, occur only here in the New Testament).
(b) The connection between ver. 8 and verses 9-20 is so awkward that it is difficult to believe that the evangelist intended the section to be a continuation of the Gospel. Thus, the subject of ver. 8 is the women, whereas Jesus is the presumed subject in ver. 9; in ver. 9 Mary Magdalene is identified even though she has been mentioned only a few lines before (15.47 and 16.1); the other women of verses 1-8 are now forgotten; the use of αναστας δε and the position of πρωτον are appropriate at the beginning of a comprehensive narrative, but they are ill-suited in a continuation of verses 1-8.3
Commentator J.R. Dummelow adds a few more:
(1) the true conclusion certainly contained a Galilean appearance (Mark 16:7, cp. 14:28), and this does not.
(2) The style is that of a bare catalogue of facts, and quite unlike St. Mark's usual wealth of graphic detail. . . .
(6) The section seems to represent not a primary tradition, such as Peter's, but quite a secondary one, and in particular to be dependent upon the conclusion of St. Matthew, and upon Luke 24:23f.2
In favor of Mark 16:9-20 being authentic, Scrivener responds to these arguments with the following dismissal:
With regard to the . . . alleged difference in style from the rest of the Gospel, I must say that the same process might be applied — and has been applied — to prove that St. Paul was not the writer of the Pastoral Epistles . . . St. John of the Apocalypse, Isaiah and Zechariah of portions of those prophecies that bear their names. Every one used to literary composition may detect, if he will, such minute variations . . . either in his own writings, or in those of the authors he is most familiar with[!]4
Scrivener is highlighting a major weakness in "stylistic clues": they tend to be easily bent to the will of the interpreter! This has been my experience as well, and so I tend to place less weight on such arguments (though they can be very telling in some cases.) But for example:
Metzger argues that it is strange for Mark to mention Mary Magdalene by name in 16:9 since she had already been named in 16:1, yet note that she is also explicitly named in 15:40 and 15:47 (one verse prior to 16:1)! With all due respect, this is a silly argument, and reflects a haste to justify a preconceived notion.
There are 17 words in Mark 16:9-20 that are not previously used in Mark's Gospel, which Metzger and others use as proof that Mark didn't write it. Yet, if you take the previous 12 verses in Mark (15:44-16:8) and apply the same criteria you find that this section also uses 17 new words that did not previously appear in Mark!7
The arguments against authenticity seem to be rooted in the writings of Eusebius,4 who, ironically, seems to have favored the inclusion of these verses.5 Regardless, both external evidence and internal structural evidence weigh in favor of the authenticity of Mark 16:9-20. Some have argued that internal stylistic evidence points to the contrary, but upon closer examination, these claims seem to be weak at best. Stylistic clues should be considered, but only with the utmost caution, given the nature and track-record of this method of analysis.
Thus, the evidence -- both internal and external -- seems to indicate that Mark 16:9-20 was, in fact, originally part of Mark's Gospel.
1: http://www.soniclight.com/constable/notes/pdf/mark.pdf, pp. 206-207
2: http://www.bible-researcher.com/endmark.html citing A Commentary on the Holy Bible, edited by J.R. Dummelow (New York: MacMillan, 1927), pages 732-33
3: http://www.bible-researcher.com/endmark.html citing Bruce Metzger, A Textual Commentary on the Greek New Testament (Stuttgart, 1971), pages 122-126
4: http://www.bible-researcher.com/endmark.html citing F.H.A. Scrivener, A Plain Introduction to the Criticism of the New Testament, fourth ed. (London: George Bell and Sons, 1894), volume 2, pp. 337-344
5: https://www.apologeticspress.org/apcontent.aspx?category=13&article=704 citing Aland, Kurt and Barbara Aland (1987), The Text of the New Testament (Grand Rapids, MI: Eerdmans), page 287
7: https://www.apologeticspress.org/apcontent.aspx?category=13&article=704 citing Broadus, John A. (1869), “Exegetical Studies,” The Baptist Quarterly, 3:355-362, July.