IMO the accuracy potential of any firearm is governed by the same set of factors no matter what platform it is built from. So if your technique can compensate for the longish lock time, TC Contenders and Encores can be as precise as any other platform, in most circumstances. That said the required shooter technique is the difference in some cases. An example would be serious benchrest competition where all else being equal by design the TC's do not as easily allow the exact same repeatability of technique as say a bolt rifle does. A product of having to take the TC "off line" to reload. So the entire "weld" between firearm to rest and shooter to firearm have to be reestablished for each shot. That's not as conducive to repeatability. Even so with long practice of technique, some TC shooters will give even the race rifles a run for the money (and that includes TC handguns).
We can all relate the "I" things for our experiences, so here's mine. I've had plenty of TC's that would shooting in the 2's and 3's consistently, both handgun and carbine lengths, one in the 1's. But many were premium barrels with match chambers shooting tuned loads to go along with very long experience honing my techniques. Most factory barrels will do MOA or better with tuned handloads as well, and some of them will shoot as small as the custom barrels.
...are the Thompson single shots more of a hunting gun, OK for getting a deer at 100 yards but not a tack driver at 300?
I've had many Contender barrels that were tack drivers at far longer ranges than 300 yards. 300 is just a good stretch of the legs IOW, two to three or more times that is the workout. The only thing you give up hunting with a TC is the speed of follow up shots. Like when calling coyotes alone and 6 or 8 come in all at once (been there). Many folks with good field shooting technique don't need a follow up with a TC hunting - some can do follow ups almost as fast as with a bolter if needed.
As for trigger jobs on TC's, Jim Henry (Barstow, CA) has always been the man, bar none. And he includes a complete frame teardown, inspect/clean and re-lube for less money than the others charge for just the TJ (way less than some of them). Also be aware that some TJ's are done by methods that are fast for the smith, but very unsafe sooner or later.