If your counter against that is that 0 will never become 1 no matter how many you add, then that just proves 'infinite' correct. If it ever could, it wouldn't be infinite...

You're confusing infinity for unreal numbers. Infinity and negative infinity are not real numbers, but not all unreal numbers are infinity or negative infinity.

If you're strictly adding zeros, then adding infinite zeros nets you zero. If adding zero once didn't change the result, then adding it infinite times won't either. If you need to add enough zeros to get to 1, that number doesn't exist - but that doesn't mean that it's infinity, it means that there's no solution. Infinity is a placeholder for "larger a real number than you can imagine", but when you multiply that by zero, the magnitude of infinity is a moot point because you have zero infinities.

In calculus if you're curious, you're usually not strictly adding zero itself like above but instead adding values that approach zero. In that case, 0*infinity really "a very small number times a really big number", and that is called an "indeterminate form". In that case you may try rearranging it to solve