Answer updated after reviewing the episode transcript.
In keeping with <PERSON> and <PERSON>'s existential themes of nihilism and absurdism, <PERSON> acknowledges a nonsensical limit to his seemingly limitless abilities.
<PERSON> specifically mentions in this episode that R&M occurs in an infinite multiverse:
There's an infinite number of realities, <PERSON>, 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, <PERSON>. 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 <PERSON>'s exact criteria are met. If <PERSON> is speaking figuratively here (assigning the value of "infinity" to an enormous [but limited] number of universes as a simplification for <PERSON>), my original answer (below) may still apply, but I'm inclined to take <PERSON>'s words as literal and demonstrative of the show's philosophy.
Before this scene, <PERSON> says:
I do have one emergency solution that I can use that'll kind of put everything back to normal, relatively speaking. Here, <PERSON>, put this on while I do a little bit of scouting.
<PERSON> has to search for these universes where he's solved their problems and he and <PERSON> die (so "original" <PERSON> and <PERSON> can easily assume their identities). 663
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, <PERSON>'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 <PERSON> and <PERSON> 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 <PERSON> and <PERSON> occurs in an infinite multiverse containing a limited number of universes that feature <PERSON> and <PERSON>. (This is what <PERSON>'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 <PERSON> expresses early in the episode, our understanding is limited and everything is up for interpretation:
Okay, well, sometimes, science is more art than science, <PERSON>. 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 <PERSON>’s frustration to curb the ‘infinite’ part of ‘infinite realities.’ (Truly infinite realities would be limitless, and <PERSON>’s concern that he and <PERSON> might “run out” of suitable replacement worlds would be misguided.)
A limited multiverse is more sensible in-context: In this scenario, <PERSON> is aware of the available replacement worlds and recognizes that a vast majority of other universes are not habitable long-term for him and <PERSON> (e.g. the <PERSON> world).
These suitable replacements feature minimal variance from <PERSON>’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.) 283