The Schur multiplier $H^2(G;{\mathbb C}^\times) \cong H^3(G;{\mathbb Z})$ of a finite group is a product of its $p$-primary parts
$$H^3(G;{\mathbb Z}) = \oplus_{ p | |G|} H^3(G;{\mathbb Z}_{(p)})$$
as is seen using the transfer. The $p$-primary part $H^3(G;{\mathbb Z}_{(p)})$ depends only of the $p$-local structure in $G$ i.e., the Sylow $p$-subgroup $S$ and information about how the subgroups of $S$ become conjugate or "fused" in $G$. (This data is also called the $p$-fusion system of $G$.)
More precisely, the Cartan-Eilenberg stable elements formula says that
$$H^3(G;{\mathbb Z}_{(p)}) = \{ x \in H^3(S;{\mathbb Z}_{(p)})^{N_G(S)/C_G(S)} |res^S_V(x) \in H^3(V;{\mathbb Z}_{(p)})^{N_G(V)/C_G(V)}, V < S\}$$
One in fact only needs to check restriction to certain V above. E.g., if S is abelian the formula can be simplified to $H^3(G;{\mathbb Z}_{(p)}) = H^3(S;{\mathbb Z}_{(p)})^{N_G(S)/C_G(S)}$ by an old theorem of Swan. (The superscript means taking invariants.) See e.g. section 10 of my paper linked HERE for some references.
Note that the fact that one only need primes p where G has non-cyclic Sylow $p$-subgroup follows from this formula, since $H^3(C_n;{\mathbb Z}_{(p)}) = 0$.
However, as Geoff Robinson remarks, the group $H^3(S;{\mathbb Z}_{(p)})$ can itself get fairly large as the $p$-rank of $S$ grows. However, $p$-fusion tends to save the day. The heuristics is:
Simple groups have, by virtue of simplicity, complicated $p$-fusion, which by the above formula tends to make $H^3(G;{\mathbb Z}_{(p)})$ small.
i.e., it becomes harder and harder to become invariant (or "stable") in the stable elements formula the more $p$-fusion there is. E.g., consider $M_{22} < M_{23}$ of index 23: $M_{22}$ has Schur multiplier of order 12 (one of the large ones!). However, the additional 2- and 3-fusion in $M_{23}$ makes its Schur multiplier trivial. Likewise $A_6$ has Schur multiplier of order 6, as Geoff alluded to, but the extra 3-fusion in $S_6$ cuts it down to order 2.
OK, as Geoff and others remarked, it is probably going to be hard to get sharp estimates without the classification of finite simple groups. But $p$-fusion may give an idea why its not so crazy to expect that they are "fairly small" compared to what one would expect from just looking at $|G|$...
There are really two separate questions that you seem to be conflating here.
The first is how to state the CFSG in a way that could be mechanically formalized. The second is how to state the CFSG that adequately reflects how human mathematicians think about it.
For the former question, one straightforward possibility for the sporadic groups, since we know their orders, is simply to state something like, "There exists a unique simple group, not in one of the aforementioned families, of each of the following orders: 7920, 95040," etc. This is the barest possible statement that could count as a classification theorem, and for a computer, it provides (in principle) enough information to reconstruct the groups in question.
For the second question, though, there's no sharp boundary demarcating where the classification theorem ends and the detailed study of the properties of the sporadic groups begins. There's also no canonical way of describing a particular group of interest in a way that satisfies a human that he or she now "knows what the group is." But there's nothing unique about group theory here. Any sufficiently large and complicated mathematical object is going to suffer from this problem. There will be some bare-minimum way of referring to it that in principle picks it out from the amorphous universe of all mathematical objects but that fails to answer basic questions about it. There will be a continuum of theorems that answer other basic questions, shading off into questions that we can't answer. It is a matter of opinion how many questions we have to be able to answer before we can claim to have "adequately described" the object.
Best Answer
Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html), $G$ has an abelian normal subgroup of index at most $(d+1)!$, which must be trivial since $G$ is simple, so $|G| \leq (d+1)!$. Rearranging, $d \gtrsim \log n / \log\log n$.
Collins's work builds on work of Weisfeiler that I think was unfinished by the time of his disappearance.
Edit: Specializing to simple $G$ actually reduces Collins's paper to a reference to a reference to the paper of Seitz and Zalesskii (https://www.sciencedirect.com/science/article/pii/S0021869383711324?via%3Dihub) mentioned by David Craven in the comments, so that's really the heart of the matter. We thereby get the slightly stronger bound $n \leq (d+1)!/2$ (for sufficiently large $d$). Apart from alternating groups I think you can read out a much stronger bound like $n \leq \exp O((\log d)^2)$, or $d \geq \exp \Omega( (\log n)^{1/2})$ (I am guessing the next worst case is $\mathrm{SL}_n(2)$).