Vail wanted the jury to see the tape, but he was worried that it might backfire if he were the one to present it.
The tape revealed that:
the archbishop was a long-time serial child molester
That fact is very helpful for his client. And the tape is an optimally graphic and powerful proof of that fact, which makes it literally the ideal piece of trial evidence.
The drawback is that the content is not merely indecent, but pretty universally repugnant (and actually might be illegal in itself). So it's in very poor taste to expose 12 unprepared jurors to that kind of material. And it absolutely destroys the victim, who was a beloved public figure. So there are some strong reasons why a lawyer might, as a matter of decorum, find a drier, less visceral way to establish that fact. But Vail deliberately chose to do it in the bluntest, loudest way, because he (quite reasonably) believed that doing so was guaranteed to make the jury more sympathetic to his client. His goal was to exploit the jury's emotions, and he goes hog wild when presented with the opportunity par excellance.
So, he tries to manipulate opposing counsel into doing it for him, so that it won't look so much like the accused is slandering the victim out of desperation to avoid conviction.
But opposing counsel is smart enough to wriggle out of this trap: she drags his accomplice onto the witness stand where she can try to force him to reveal the cynical calculation behind Vail's maneuver.
Why does Vail not object to this counter-attack?
It's clearly a calculated risk. On the one hand, he succeeds in getting the tape admitted. On the other hand, he's (probably) about to be exposed before the jury as a supremely manipulative person. What he doesn't know is: how will opposing counsel spin his decision to block the testimony?
My read is that he lets the testimony proceed because it's the devil he knows. Opposing counsel was, to his surprise, agile enough to turn his first trick back onto him. He fears she'll spin his refusal in a way that is even more damaging than the truth. And he has to make this calculation fast, in front of the jury.
So he sticks with the truth, which is ultimately the most defensible course anyway. Maybe the jury won't mind that he's such a calculating mercenary if they at least think he's honest.
EDIT: I should also add that Vail knows Thomas will try hard to make his testimony helpful to Vail and unhelpful to opposing counsel. IIRC, it's clear Vail & Thomas have a longstanding professional relationship and so are real allies, and that Thomas is a professional who is comfortable and competent in a courtroom setting in a way that many civilians and laypeople are not. And Vail can still object during the examination.
MOAR EDIT: I also think Primal Fear was specifically concerned with the role of the reputations of high-profile defense attorneys. The historical context matters: the movie came about about 4 years after the O.J. trial. That was a time when it seemed the whole country knew everything about the lawyers and courtroom staff. Kids in middle school had opinions about attorney ethics, I sh*t you not. The movie was based on a book published in 1993, the year after the trial. Timing-wise, Primal Fear was part of what I think was a second generation of courtroom movies triggered by the O.J. saga.
So, it was not a surprise to the average American when this movie went to so much trouble to put a high-powered defense lawyer in a situation where he must acknowledge on the record that he is, himself, a cunning, hyper-manipulative mercenary who cynically games-out the jury's psychology for whoever has the cash to pay for his ostentatious lifestyle.
Part of what made Primal Fear so good was that it had a novel and plausible way to motivate that situation. (At least, that's how it seemed to me back then, when I basically lived on movie previews.)
