quantum field theory - How can dimensional regularization "analytically continue" from a discrete set?

The procedure of dimensional regularization for UV-divergent integrals is generally described as first evaluating the integral in dimensions low enough for it to converge, then "analytically continuing" this result in the number of dimensions $d$. I don't understand how this could possibly work conceptually, because a d-dimensional integral $I_d$ is only defined when $d$ is an integer greater than or equal to 1, so the domain of $I_d$ is discrete, and there's no way to analytically continue a function defined on a discrete set.

For example, in Srednicki's QFT book, the key equation from which all the dim reg results come is (pg. 101) "... the area $\Omega_d$ of the unit sphere in $d$ dimensions ... is $\Omega_d = \frac{2 \pi ^{d/2}}{\Gamma \left( \frac{d}{2} \right) };$ (14.23)". But this is highly misleading at best. The area of the unit sphere in $d$ dimensions is $\frac{2 \pi^{d/2}}{\left( \frac{d}{2} - 1 \right) !}$ if $d$ is even and $\geq 2$, it is $\frac{2^d \pi^\frac{d-1}{2} \left( \frac{d-1}{2} \right)! }{(d-1)!}$ if $d$ is odd and $\geq 1$, and it is nothing at all if $d$ is not a positive integer. These formulas agree with Srednicki's when $d$ is a positive integer, but they avoid giving the misleading impression that there is a natural value to assign to $\Omega_d$ when it isn't.

Beyond purely mathematical objections, there's a practical ambiguity in this framework - how do you interpolate the factorial function to the complex plane? Srednicki chooses to do so via the Euler gamma function without any explanation. But there are other possible interpolations which seem equally natural - for example, the Hadamard gamma function or Luschny's factorial function. (See http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html for more examples.) Why not use those?

In fact, these two alternative functions are both analytic everywhere, so you can't use them to extract the integral's pole structure, which you need in order to cancel the UV infinities. To me, this suggests that the final results of dim reg might be highly dependent on your choice of interpolation scheme, therefore requiring a justification for using the Euler gamma function. Could we prove to a dim reg skeptic that all results for physical observables are independent of the interpolation scheme? (Note that this is a stronger requirement than showing they are independent of the fictitious mass parameter $\tilde{\mu}$.)

(I know that the Bohr-Mollerup theorem shows that the Euler gamma function uniquely has certain "nice" properties, but I don't see why those properties are helpful for doing dim reg.)

I'm not looking for a hyper-technical treatment of dim reg, just a conceptual picture of what it even means to analytically continue a function from the discrete set of positive integers.

Edit: It appears that the details of exactly which field-theory results do and do not depend on the choice of regularization scheme are not well-understood; see this paper for one discussion.