Q. What are the characteristics of theorems that seem to invite (or possess) several or even many distinct proofs?

What I have in mind are examples such as these:

• Proofs that there are infinitely many primes.
• Proofs of irrationality of $\sqrt{2}$.
• Pythagorean Theorem.
• Twenty Proofs of Euler's Formula: V-E+F=2 (David Eppstein's collection).

The last is particularly striking to me, as it took some time for an accurate proof to emerge.¹ I'm sure there are many other more modern examples; suggestions welcomed.

Is there some characteristic of these theorems that lend themselves to often rather distinct proofs? Or is it where within the network of mathematical connections these theorems reside? Or is it that these theorems are so useful that researchers keep inventing new proof approaches?