Let me make some comments on the polygonal number theorem along the theme that things are less thrilling for $k \gt 4$.
Triangular and square numbers are pleasing to represent as dot patterns, after that $k$-gonal numbers are less attractive.
Around 1796 Gauss proved that every integer is a sum of 3 triangular numbers. This is sometimes called the Eureka Theorem because Gauss was quite pleased with the result.
Evidently, Cauchy showed (around 1813) that from that one can go to the general polygonal theorem that every integer $n$ can be represented as a sum of $k$ $k$-gonal numbers.
Tables due to Peppin and Dickson show how to represent $n \lt 120k-240$ as a sum of at most $k$ $k$-gonal` numbers.
A nice paper by Melyvn Nathanson uses results 2 & 4 to show in a few pages that
a. for $k \ge 5$ and $n \ge 120k-240$, $n$ can be written as a sum of $k-1$ $k$-gonal numbers of which at most four are different than $0$ or $1$.
b. For fixed $k$ every large enough odd $n$ is a sum of four $k$-gonal numbers and every large enough even $n$ is a sum of five $k$-gonal numbers one of which is either $0$ or $1$.
So it seems possible to me that taking 2 & 4 as given, it might be possible to express at least some of the steps of a transition to the general polygonal theorem in a somewhat geometric way, however if this hypothetical thing is possible it would probably not be a simplification giving insight as much as a somewhat more complicated path which is interesting mainly for the fact that it can be done at all.
I would think of an insight giving proof as one which also showed how to represent $n$. The proofs I know of show that there must be a representation but don't tell you how to find it. Changing $n$ to $n+1$ and/or $k$ to $k+1$ can radically change the representation.