Maybe a better way to think of it is not so much as strict mathematical terms since infinites usually have more nuanced meaning in reality than just "undefined". In this case, it just means that Vs needs to supply no current (or rather, it's current is always forced to zero by something else). So no matter how much current Vs tries to supply, it can't supply it.
The simplest way of thinking about infinite as just a very large number is not very accurate and only works for simple cases. It breaks down when you get to more advanced stuff which is why it's not really making sense here. In reality, infinite isn't a large number as much as the concept of something constantly getting larger without ever being a set value. This means that you can perform math operations on it against another "number" that either increases equal to, faster, or slower than that particular infinite. This nuance is what lets you come up with real-world results that make sense when working with infinite. If we're talking in terms of pure math, stuff like 0/0 or zero x infinite is undefined, but not really. You can kind of manipulate things and play tricks to get an answer out of it that reflects what happens in the real world (not always, but that's where the limits of human understanding are).
In not-so-strict hand-wavey pure math terms thinking about it in reverse: If the current gain is actually infinite but the voltage across Ri is finite, then what must the current supplied by Vs be? Well, it certainly can't be non-zero because you don't have an infinite voltage. It HAS to be zero because in the realm of crazy math zero multipled by the right kind of infnite perfectly counteract each other to become a finite number (BTW zero x infinite does not equal to zero in general in case you were not aware. In general it's undefined because the process of zeroing is up against infinite's process of increasing and without more information we can't decide which one is winning).
But what does this really mean? Well, remember when I said infinite was not a fixed number but a process of continually increasing so you could never catch up to it? There are many ways you can do this such as the rate of increase. In this specific case, this sort of infinite can sort of be thought of as "something intelligent" is constantly adjusting itself to always just barely stay ahead of you. That is infinite in the sense that you can never reach the end, but neither is it running on forever. It's a real finite value that you just can't catch up to because of your frame of reference (an outside observer that the infinite is not staying just ahead of can just go in and measure it just fine and it would be a value). So the gain of infinite is not really a fixed gain that is an infinite number, but a gain that is actually a finite number that is changing and constantly being adjusted to be "just enough" to force the Iin to be zero. In that way, everything stays defined but at the same time Vs can never overcome the obstacle of pushing current ("catching up" to the infinite gain which is actually just a finite gain that's always one step ahead of it).
People write PhD thesis on this stuff so what I told you is about all I know. I don't know if this answered your question or not but maybe it will help you approach strange scenarios like this with a more flexible mindset.
Of course, in reality there are real world limits to this circuit like the power supply voltages so a result of infinite would mean "as much as it possibly can" rather than something as nebulous as "undefined". It's just that you used approximations in your equations rather than entering all the real-world nitty gritty data that would have given you a more specific answer, so you instead you ended up with infinite as an approximation to a very large real-world number.
