Honestly I think it’s misleading to describe it as being “defined” as 1, precisely because it makes it sounds like someone was trying to squeeze the definition into a convenient shape.

I say, rather, that it naturally turns out to be that way because of the nature of the sequence. You can’t really choose anything else

X^0 and 0! aren’t actually special cases though, you can reach them logically from things which are obvious.

For X^0: you can get from X^(n) to X^(n-1) by dividing by X. That works for all n, so we can say for example that 2³ is 2⁴/2, which is 16/2 which is 8. Similarly, 2¹/2 is 2⁰, but it’s also obviously 1.

The argument for 0! is basically the same. 3! is 1x2x3, and to go to 2! you divide it by 3. You can go from 1! to 0! by dividing 1 by 1.

In both cases the only thing which is special about 1 is that any number divided by itself is 1, just like any number subtracted from itself is 0