Fucking temporal mechanics.
Okay so my current running theory is that OTL Time Trapper is Mandated Paul, and this entire thing is a closed temporal loop.
From our perspective as the readers, we hadn't seen any indication of Paul ever meeting whoever Time Trapper was supposed to be, and yet he clearly recognized him and appeared to have previously interacted with him. But even though everything in the story is from Paul's perspective, nobody has any indication of that meeting having happened- everyone's just assumed it happened offscreen.
What if, instead, it hadn't happened because we haven't seen it from Paul's perspective yet?
Paul recognizes Time Trapper from a previous meeting. Time Trapper has no recollection of this meeting. He throws Paul back in time. Then he tracks down when the meeting was supposed to occur, figures out nobody's there to meet Paul, and goes and meets him in order to secure his own personal timeline.
Paul gets flung back in time. Now this can go one of two ways. Either he arrives just in time for his timeline to get overwritten by Savage's changes, or he doesn't arrive at all. If it's the latter, this will be important later.
Truggs arrives, timeline goes on, Mandated Paul (Who arrived from Earth Prime and went through all the shenanigans he did in the new timeline) finds out about all this stuff, confronts Savage, and goes back to the night of stone knives to save his timeline along with Truggs. If OTL Paul didn't arrive at the present and get overwritten, he arrives at the same point in time- the Night of Stone Knives, and runs into his alternate timeline self at a plot convenient point to help. Truggs tries to get the original timeline to reassert itself, they all fight Savage, and ultimately the mandated timeline, or at least most of it is averted in some way that Mandated Paul can't recover from. He fucks off into the greater universe, while OTL Paul and Truggs go home via stasis the long way around.
Mandated Paul spends a bunch of time researching temporal technology because his goal is to still get his timeline back. He's functionally ageless thanks to his ring, and eventually assumes the Time Trapper identity, studying the Legion of Superheroes' collapsing timeline and theorizing that the research based on it might be enough to figure out who to re-assume his own timeline.
But when Paul recognizes him from a previous meeting, he realizes that this is the setup for his own timeline, and goes to ensure it works out (because the alternative is a future where Savage wins). This also leads to setup for I assume some kind of plan to save his timeline. Nu-52 never happened in this shard, so it's entirely feasible that with enough energy, a time traveler could shunt a timeline off into a new world line. I assume that's what Time Trapper Paul's goal is here.
Savage is just a patsy for TTPaul's Xanatos gambit to set everything up so the Mandated universe can be split off into his own distinct timeline. All the shit he's up to is leading up to a means to generate enough energy to stabilize the timeline, and that's why Time Trapper was studying the Legion's timeline. Same deal.