There are two broad categories of "gun does not fire" issues.
#1) the empty gun does not go "click" when the trigger is pulled
#2) the gun goes click when empty, but does not go bang when loaded.
While the end result is the same (no shot fired) the reason for this, and the parts involved can be quite different between case#1 and #2.
Take the gun to a gunsmith, let them figure out what is wrong. That's one of the things they do.
Flite Kings were regarded as solid "working class" gun, fine for function, with finish better than bargain guns but not quite up to the big name makers standards. My brother had a 20ga when we were teens. It worked fine.
There are some nicely finished Flite Kings around, but it was a more "economy" class gun than Winchesters and Remingtons.
All else being equal (and it almost never is) bigger bullets tend to work better.