It may well be that an intuitive picture for the logarithmic correction could already be found in the paper of Ashoke Sen which was the starting point of the question (Here is the link again: 1205.0971. Maybe the fact that the argument was somewhat hidden page 20 (§2.5 Higher loop contributions) is the reason why it was not obvious at first sight. But as emphasized later by Jeon and Lal in the introduction of 1707.04208 pages 3:
The main reason why the log term is an important contribution to the
microscopic formula is that it is a genuinely quantum correction
to the Bekenstein-Hawking formula determined completely from
one-loop fluctuations of massless fields,
which essentially constitute the IR data of the black hole. [...]
The log term may be regarded as an IR probe of the microscopic theory,
in the sense that any putative microscopic description of the black hole
must correctly reproduce not only the leading Bekenstein-Hawking area law,
but also the log correction to it.
Now, Sen's argument is simply based on naive power counting:
in $D$ dimensions, the $\ell$-loop contribution to the free energy, given by a typical Feynman (vacuum) graph, should scale as
$$\left(\frac{\ell_P}{a}\right)^{(D-2)(\ell-1)}\!\!\int^{a/\sqrt{\epsilon}}\!\!\!\!d^{D\ell}\tilde k\;\tilde k^{2-2\ell}F(\tilde k),$$
where $a$ is the black hole size parameter related to the horizon area $A$ via $A\sim a^{D-2}$ and $\ell_P$ is the Planck length related to Newton's constant $G$ via $G\sim\ell_P^{D-2}$. I'm keeping here the notations of the question for the area and Newton's, and the ones of Sen for the rest. The integration variable $\tilde k$ is related to the loop momenta $k$ via $\tilde k=ka$. $\epsilon$ is an $a$-independent ultraviolet cut-off (of the order of $\ell_P^2$ in an ultraviolet regulated theory). The multiplicative function $F(\tilde k)$ encodes the modifications (from their form in flat space-time background) of the various propagators and vertices carrying momenta $k$ in the presence of the black hole. So $F(\tilde k)$ approaches $1$ for large values of $ka$, as we expect to recover the propagators and vertices in flat space-time background for large momenta.
First consider the case where all loop momenta are of the same order. As $F(\tilde k)\rightarrow 1$, we can expand the function in a power serie in $1/\tilde k$ for large $\tilde k$. A $\ln(a/\sqrt{\epsilon})$ term will come out of the integration from the $\tilde k^{2\ell-2-D\ell}$ contribution and after multiplication by the $a$-dependent prefactor it will look like
$$\left(\frac{1}{a}\right)^{(D-2)(\ell-1)}\ln a$$
which is highly suppressed in the large $a$ limit unless $\ell = 1$, as advertised.
Furthermore, the possibility that a subset of the loop momenta is smaller than the rest, was analysed by Sen. The effect of the hard loops can be regarded as renormalization of the vertices and propagators of the soft part of the graph. We can conclude with Sen that as long as the renormalization doesn't change the low energy effective action, that is, as long as the massless particles are kept massless and minimal coupling to gravity remains minimal, these contributions do not change the logarithmic corrections to the black hole entropy. At last, the higher derivative couplings that could be generated by renormalization effects will give additional powers of $\ell_P/a$, making the coefficient of the logarithmic term even more suppressed than what has been argued before.
Therefore, we can say that the one-loop correction to the Bekenstein-Hawking law is universal: it depend only on the massless spectrum and is insensitive to the UV completion of the theory. To put it differently, the basic reason for this is that the effects of a massive particle can be accounted for by integrating it out, which generates higher derivative terms in the effective action, which in turn lead to corrections to the entropy which are suppressed by inverse powers of $a$ and cannot contribute.
If you are still not fully convinced about the canonical status of the "genuinely quantum" log-correction, let me first emphasize the central role of the heat kernel method in Sen's computation (see e.g. §2.2 p. 10), which is a very convenient tool for computing one-loop divergences and studying quantum anomalies (see Vassilevich for a general view). Along these lines, we may be even more convincing by using the modern approach which recasts the computation of the entropy of black hole in terms of entanglement entropy. More precisely, following Solodukhin, Ro Jefferson has written a very
informative blog post on the topic which gives, via the replica trick, the two contributions to the thermodynamic entropy of the (Schwartzschild) black hole: a classical (because it represents the tree-level contribution to the path integral) gravitational entropy, the famous Bekenstein-Hawking area law, and the entanglement entropy which is the one-loop quantum correction. That is, if we were to restore the Planck constant, the area law would come with a $1/\hbar$, and the entanglement part would be of order $\hbar^0$.
