Answer updated after reviewing the episode transcript.
In keeping with Rick and Morty's existential themes of nihilism and absurdism, Rick acknowledges a nonsensical limit to his seemingly limitless abilities.
Rick specifically mentions in this episode that R&M occurs in an infinite multiverse:
There's an infinite number of realities, Morty, and in a few dozen of those, I got lucky and turned everything back to normal. I just had to find one of those realities in which we also happen to both die around this time. Now we can just slip into the place of our dead selves in this reality and everything will be fine. We're not skipping a beat, Morty. Now, help me with these bodies.
In itself, this is a paradox: If there were infinite realities, there would be a limitless number of realities wherein Rick's exact criteria are met. If Rick is speaking figuratively here (assigning the value of "infinity" to an enormous [but limited] number of universes as a simplification for Morty), my original answer (below) may still apply, but I'm inclined to take Rick's words as literal and demonstrative of the show's philosophy.
Before this scene, Rick says:
I do have one emergency solution that I can use that'll kind of put everything back to normal, relatively speaking. Here, Morty, put this on while I do a little bit of scouting.
Rick has to search for these universes where he's solved their problems and he and Morty die (so "original" Rick and Morty can easily assume their identities). Regardless of your theoretical approach, this creates all kinds of problems in the context of infinite universes (essentially, infinite realities beget infinite solutions). Moreover, as you mention, Rick's words relay concern for the future in "Morty's Mind Blowers" ("I said that we can only do that a couple of times!"), which doesn't make sense unless the number of universes where Rick and Morty exist is limited, again bringing us to my original 'limited multiverse' answer.
It's possible (maybe, I'm not an expert on any of this scientific theory) that Rick and Morty occurs in an infinite multiverse containing a limited number of universes that feature Rick and Morty. (This is what Rick's own words suggest, by my logic.) This doesn't really make sense with respect to the meaning of "infinite," but Rick and Morty thrives on these paradoxes. As Rick expresses early in the episode, our understanding is limited and everything is up for interpretation:
Okay, well, sometimes, science is more art than science, Morty. A lot of people don't get that.
Original answer – limited multiverse scenario
It’s partly for comedic effect, but it also uses the subtext of Rick’s frustration to curb the ‘infinite’ part of ‘infinite realities.’ (Truly infinite realities would be limitless, and Rick’s concern that he and Morty might “run out” of suitable replacement worlds would be misguided.)
A limited multiverse is more sensible in-context: In this scenario, Rick is aware of the available replacement worlds and recognizes that a vast majority of other universes are not habitable long-term for him and Morty (e.g. the Cronenberg world).
These suitable replacements feature minimal variance from Morty’s home universe and are limited in quantity as a function of basic statistics: Imagine “universe characteristics” follow a Poisson distribution – any given set of desired characteristics in the overall group (when that characteristic group has as many parameters as a universe) is going to be statistically unlikely and occupy a comparatively small solution space. (Here, that small solution space is the “three or four” universes out of billions+ options.)