Confused on application of Anti-Aliasing Filter I have a question about the use of anti-aliasing filters prior to sampling a finite time signal and if they actually help anything.

The point of the anti-aliasing filter is to remove high-frequency components to reduce aliasing. It's know a signal cannot be both band-limited and time-limited thus time limited signals are first low-passed to "reducing aliasing",

This sounds good but it seems to create its own problems in so far as it will create infinite(or near infinite) duration output as a response to a finite input.

We can't sample forever, no matter what the signals samples will be finite in quantity. So aren't we creating as big of a problem as we are solving?
Math and Physics are not the same thing. They can be very close but infinity is hard to realize.  Going from analog to digital, a more accurate description is an average of the waveform over a small time window and the average is in turn quantized. Filters don’t have infinite stop bands but 80 dB is usually more than good enough.  There is always a small error, but one can bound it.