So that line is interesting because it isn't necessary limiting the number of available timelines to jump but rather the practical limits on finding and jumping to such a timeline. There are a few factors at play here:

First the dimension they jump to has to have the following:

1. The same general history up until this point (otherwise they won't fit in) imagine how hard it would be for Rick and Morty to blend in on the dimension that the Cowboy versions of them are from?
2. Both Rick and Morty have to have recently died and not yet been discovered by anyone of significance. If they were off world this is easier and their window is much larger. But if the death happens on Earth then they have a narrow shot to hit.

Regardless of the size of the window even finding a universe in the multiverse that up until this moment has been roughly the same minus one major fuckup by Rick in which Rick and Morty both then diverge and die shortly after avoiding the fuckup is quite the needle in the haystack to find.

This is compounded by the fact that for every Rick who didn't fuck things up but then died thereafter there is just as likely to be a Rick who did fuck things up and not die and is now looking to jump dimensions. So we can see a third requirement on the dimension they are looking for: That no other Rick is jumping into that dimension before our Rick does.

Now given these incredibly stringent requirements we can see two things:

1. Scanning the multiverse to find such an idea dimension would be very time consuming.
2. The longer time goes on the fewer dimensions there will be which satisfy these requirements as divergence happens. A process which is only hastened every time Rick makes a big mistake and a whole lot of Ricks all need to find new universes to live in.

Even if there were an infinite number of universes to hop to, they will still be contained inside a much larger also infinite set of all universes and thus eventually the time needed to find a suitable destination will grow until it exceeds the gain caused by hopping dimensions and eventually also exceeds the expected lifespans of the Ricks and Morties who are trying to escape.