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.