I have been thinking on this problem for at least a day now, and I thought the answer is a resounding no. It seems I am wrong, but wanted to double-check. I came across this:
And from my understanding assuming that WLOG one can take any linear subspace to be $\mathbb{R}^k$ with $k$ elements of the standard basis of $\mathbb{R}^d$, this technically means if $f\in\mathcal{H}_k(\mathbb{R}^d)$ and it is the case that the non-zero coefficients of a given $f$ tend to coincide with the elements of the basis that have their $\alpha_i\neq 0$ for some $i\leq k$, and the rest of the coefficients are zero would be an example of such a function. So the answer is a resounding yes, I guess?
UPDATE: My solution definitely is not right. In fact what I call WLOG is not at all WLOG. But I would appreciate any guidance/help towards whether the answer to this question is positive or negative
What seems to be the issue? The result you quote explains that the RKHS has a basis of square-exponentially weighted polynomials. You can take any set $V$ defined by a finite number of polynomial equations and the whole bunch of polynomial functions will vanish on that set (the ideal of the corresponding affine variety); multiplying them by squared-exponential weight won't change the vanishing. So you will have a lot of functions in the RKHS vanishing on $V$ (and more of them when you take the completion, but we don't even need to go there for what you are asking).