Consider this related case: if you evaluate a limit and you get 0/0, you recognize that you need to do more work to find the actual limit. It could be 1, -1, 0, infinite, etc. depending on how you reached it (say, sin(x)/x versus sin(x)/x^2). The issue is not the continuity of x/x; the issue is whether setting a convention for 0/0 would give you the right value for a limit. Since it doesn't always, we call it "indeterminate".

Similarly, if you're evaluating a limit and you get 0^0 you need to do more work. You can't just stop and say "oh, that's 1". It depends on what function you used to get there -- x^x will give you a different answer from ( e^(-1/x) )^x. Again, it has nothing to do with the continuity of exponentiation. The issue is whether the convention of 0^0=1 is correct in the specific part of mathematics you're working in.

The same argument can be made if you're working in the hyperreals, or if you're working with field axioms -- the convention 0^0 doesn't work in that context.

Please, by all means, use the convention 0^0=1 when it's appropriate. But understand that it's not always appropriate. Not every mathematician works in the particular subschool that you do; not every mathematician is going to find your convention appropriate.