Here are some of the reasons why Christians who were lying or speculating about the authorship of the second gospel probably wouldn't have come up with Mark as the author:
"Moreover, in almost every early tradition that we know, both within and beyond the New Testament, 'Mark' cuts a decidedly second- or third-rate figure. Among the fascinating characteristics of the early traditions about Mark are their proliferation and oddity: relative to their references to the other Evangelists and Gospels, patristic texts seem to discuss Mark more yet use his Gospel less. Furthermore, in their comments about the Evangelist, the majority seem noticeably awkward, apologetic, and sometimes even pejorative….Less important than adjudicating among these interpretations of kolobodaktylos [a title applied to Mark by patristic sources], all of which are unavoidably speculative, is observing that which they hold in common: the nuance of diminished integrity, whether in regard to physical or moral capacities. A term like kolobodaktylos, with its connotations of deformity or cowardice, probably would have registered among listeners or readers in Mediterranean antiquity as a slur or (at best) as a lament, not as a compliment….First, for Eusebius the figure of Mark is of only minor concern, secondary to other historical preoccupations. Although the Church History has proved to be a mine of traditions about Mark and other personalities of the apostolic age, nowhere in that work does Eusebius devote a section to Mark as a leading character in his own right….On the teeming stage of Christian antiquity Mark is little more than a bit player." (C. Clifton Black, Mark: Images Of An Apostolic Interpreter [Minneapolis, Minnesota: Fortress Press, 2001], 11-2, 118, 158, 254)
So, Mark was a relatively minor figure who was remembered for some disreputable behavior and/or an embarrassing physical deformity.
You could try to lessen the weakness of Mark as a potential author by denying that he's the Mark who's criticized in Acts 13:13 and 15:36-40. But even if Mark's disreputable behavior is taken off the table, the other two problems - his minor role in early church history and his physical deformity - remain. And even if we were to conclude that the Mark of Acts 13 and 15 is some other Mark, the view of the early Christians is more important in this context, and they didn't make that distinction between two Marks. The simplest explanation for the New Testament references to an associate of the apostles named Mark is that there was one Mark rather than multiple apostolic associates who happened to have the same name. And the early Christian consensus seems to be that there was no second Mark, much less three or more. On page 160 of his book cited above, for example, Black notes that Eusebius seems to have had no concept of a second Mark. Eusebius had access to a lot of early Christian literature that's no longer extant, and he discusses belief in more than one early church leader named John (Church History, 7:25), but he seems unaware of any comparable situation with Mark. Rather than the early Christians having erred in thinking there was just one Mark, it's more likely that the speculation of more than one Mark is incorrect. It's not as though combining multiple figures into one is the only direction in which people can err:
"Ancient writers sometimes confused persons of the same name, but they also sometimes created new persons on the supposition that two persons of the same name had been confused. Thus a story was circulated that the Pythagorean diet was to be attributed to a different Pythagoras, a story which Diogenes Laertius prudently found unpersuasive. In a case not unlike John the elder versus John the apostle, some opined that Pythagoras the philosopher had a student with the same name responsible for the athletic treatises wrongly ascribed to the teacher. Distinctions demanded by divergent traditions yielded more than one heroic Heracles and more than one Dionysus." (Craig Keener, The Gospel Of John: A Commentary, Vol. 1 [Peabody, Massachusetts: Hendrickson Publishers, 2003], 97)
The Mark mentioned in Acts is closely associated with Peter, Paul, and Barnabas in that book (12:11-2, 12:25, 13:13, 15:36-40). Somebody named Mark is closely associated with those three individuals elsewhere in the New Testament as well (Colossians 4:10, 2 Timothy 4:11, Philemon 24, 1 Peter 5:13). Mark is often named in close proximity to Luke (Colossians 4:10-4, 2 Timothy 4:11, Philemon 24), which makes sense not only if just one Mark is in view, but also if this Mark wrote a gospel that Luke used as one of his sources when composing his own gospel. There's probably only one Mark in view in all of the relevant passages. So, there was one individual who had all of the weaknesses Black refers to above, yet he was unanimously named as the author of the second gospel.
Sometimes, 1 Peter 5:13 is brought up as an alleged source for the tradition, but that explanation doesn't make sense. It fails to explain why the gospel was associated with Peter to begin with. It also doesn't explain why the gospel wasn't attributed to Peter himself or a more prominent and/or less problematic disciple of Peter. Silvanus, for example, who's mentioned in 1 Peter 5:12, is spoken of more highly than Mark, doesn't have Mark's negative characteristics, and is mentioned specifically in a literary and canonical context ("Through Silvanus…I have written to you"). Even better, why not attribute the second gospel to Andrew, one of the more prominent apostles and Peter's sibling? And given how late critics often want to date 1 Peter, why is the attribution of the second gospel to Mark so early (the elder cited by Papias, etc.) and unanimous? If the second gospel was circulating for a while before 1 Peter was composed, as the later dates for 1 Peter suggest, why don't we see the attribution of the second gospel to Mark arising later and inconsistently? 1 Peter 5:13 isn't mentioned much in early patristic discussions of Mark. It's a poor explanation for the origins of the gospel's authorship attribution.