When you say there is no "immigrant problem", you are kidding yourself. There is no "immigrant problem" for you because you HAVE CAPPED YOUR POPULATION. However, if you had not capped your population you would have FPS trouble. Which is a problem. And killing immigrants would solve that problem. Clogging up your fort is a problem, killing immigrants is one solution.
That really depends on your definition of problem. The "problem" of lower frame rates with more dwarfs is of a different character - it depends what machine you're running the game on, for instance, if you play the same fortress on different machines, which I've done a few times - to the "problem" of having to manage and feed immigrants, which I believe is what most people here usually mean when discussing an "immigrant problem". I
prefer to maintain a good framerate, but it's a only a "problem" external to the running of a fortress, not internal to it.
For that matter, framerate is only somewhat loosely tied to population, and is heavily dependant on the jobs you have your dwarfs doing. There are circumstances where increasing the population may yield a higher framerate; I'm not entirely sure
how, some arcane quirk of the mechanics, but I've seen it happen.
Anyway, having said that, I believe there
is a problem of sorts with dwarf population - that the baby/child cap and population cap are separate entities. This separation means that in a fortress at or above the population cap (whether that's 20 or 200), if there are no dwarf casualties and the baby/child cap is above zero the population will inexorably rise above the population cap (as children mature into adults and open a new "space" in the baby/child cap, for a net gain of 1 dwarf each time) and will not normalise until the dwarves start dying of old age, and that
is a problem internal to the game.