That's certainly the route the novels took! Which is great if you have several pages of novel to explain thoroughly. I DM a lot though, so I prefer quick explanations without too many follow up questions. 😂 (I also think Section 31 feels out of character for Trip, but that's personal preference)
I remember the novels well. Section 31 IS out of place for Trip, no doubt. In my head they faked his death or used some alien device as a means of persuading him but in the end after one mission he told them where to go!
I'll just copy this head canon from you, all right?
Section 31 was established in Enterprise - otherwise, they might have used a "regular" Starfleet secret service, with less problematic connotations. And that could still be used, if necessary.
It was established as a regrettable piece of Malcolm's past, though. Trip takes things personally, he feels the least likely to work with them. But, he's the one who was dead, and a known character with that clandestine perspective was narratively useful. No harm in using the toys you're given.
As for the books, they handled this well. As for a possible Enterprise revival, I'd be happy with them going the Snake Plissken route: "I thought you were dead!" Trip: "If I ever find the guy who started this damn rumor, I'll stuff him in an intermix chamber...!" 😉