Not exactly sure what you are talking about when you say the "screw under the trigger" and "screwing the wheel" when trying to revove it.
The older M19 / 66 "K" frames and...I believe...some early "L" frames (586 / 686)...maybe the 6in. 686's still have it...not sure... had a small "trigger stop" behind the trigger which was screwed into a threaded hole in the frame and could be accessed by removing the sideplate and taking out the trigger. This trigger stop screw could back out under recoil and allow the trigger stop to rotate enough to prevent the trigger from moving all the way through its' arc, thus tying the action up. It can...and should be removed from duty guns, but may also be loctited in place for target guns. It has zero effect for DA shooting anyway.
At some point, S&W also changed the yoke screw (screw in sideplate that is closest to the muzzle) from a fitted, solid screw to a design using a hollow, "female" nut ...which is threaded like a screw...with a spring-loaded plunger inside to keep tension on the yoke. (For those who do not realize it, the yoke screw and the other round-headed sideplate screw on three-screw S&W's made before this change are the same EXCEPT for the fact that the yoke screw is hand fitted to the yoke.) It is not uncommon to find where they have been switched by users who then may find that the yoke, complete with cylinder, will come right off of the frame when the cylinder is opened and the muzzle pointed downward. Simple fix? Switch screws. The flat-headed screw always goes under the grips.
Should you decide to remove the yoke and cylinder for cleaning, etc. be aware that it is not uncommon to lose the spring and plunger from the later models when doing so. Yes, the gun will still shoot with the cylinder back in correctly...but reloading will be very slow...Ha!
Maybe that helps?