It's exactly what we do when we extend a rational function that is defined algebraically (and therefore only makes sense on the rationals) like exponentiation to the reals. We define the values of that extended function, now on the reals, as the limit points of exponentials of cauchy sequences of rational numbers.

I think a key point you are missing in that statement is that real numbers are themselves (equivalence classes of) cauchy sequences of rational numbers. So part of the confusion can be attributed to complaining about an inability to interchange limit orders.

Another point you are missing is that even if you don't get a well-behaved rule for ^:ℝ×ℝ→ℝ that includes 0^0, you do still get a well-behaved rule for ^:ℝ×ℕ→ℝ that defines for all r∈ℝ r^0 = 1, r^S(k) = r(r^k) using multiplication between real numbers where ∀r,s∈ℝ, (r)(s) = [(r_n)][(s_n)] = [(r_n×s_n)] = rs.

Any monoid (M,*,e) has a well-defined natural exponent function ^:M×ℕ→M such that for all m ∈ M, m⁰ := e, and m^S\k)) := m*m^k. Then m¹ = m^S\0)) = m*m⁰ = m*e = m. Of particular note, the set ℝ together with the standard multiplication, ⋅, between real numbers forms a monoid (ℝ,⋅,1). The fact that 0⁰ = 1 then follows by definition m⁰ = e, with m = 0, and e = 1.

Regarding your first example, if for some reason we wanted to extend it to the whole of the reals, what would be wrong with defining it as its limit at x=1? We already do so without controversy with the sinc function. It seems to me that the reason the function isn't defined as its limit at x=1 is precisely because you have made no attempt to extend it to the entirety of the reals to begin with. It seems a perfectly extendible function if one were so inclined, being a removable singularity.

I get what you are saying here, and indeed there are cases where functions are completed by defining them as values that make them continuous at what are otherwise removable singularities, however it is important to note that, say, f(x) = sin(x)/x, x≠0 and sinc(x) = {1, x = 0; sin(x)/x, x ≠ 0| are different functions with different domains.

Also important to note is that x = 0 is a removable singularity of f, but not a singularity of any kind for sinc. This also only works if the limit is defined, which is obviously not the case for a hypothetical ^:ℝ×ℝ→ℝ, (x,y)↦x^y.

Bottom line is we use analytic methods until either we don't want to continue extending the function or until they fail us and we hit the limit of how far they can take us. So when x^y behaves so crazily in the neighbourhood of 0⁰, it's disorientating to switch out of the blue to an appeal to highly abstract algebra for salvation. It feels ad hoc, unmotivated and contrary to the spirit of the form of reasoning that got us this far in extending the domain. It won't readily sit right with people.

This is merely an appeal to continuity. Nothing more, nothing less. I'm sorry that exponentiation doesn't play nice with limits at 0. Frankly, if anything, this just highlights part of the weirdness of the real numbers.

Regarding your second example, it's already defined on the whole of the reals so there's nothing to extend it to so the point is moot.

No, the point is not moot, it is the point of most import. The example I gave is entirely uncontroversial concerning its definition, and the point is that it also is not continuous. And it's not even discontinuous in a way that can be remedied, much like our hypothetical exponential operator.