Wikipedia says Palestinian presidential elections were indefinitely postponed by Palestinian President Mahmoud Abbas due to a Hamas-Fatah schism:

The elections took place amid a tense atmosphere between Fatah and Hamas over the postponement of more major elections that were also scheduled for 2021. It was also reported that voters were angry with the postponement of the presidential election and the legislative election by President Mahmoud Abbas.
https://en.wikipedia.org/wiki/2021%E2%80%9322_Palestinian_local_elections

Israel both refuses to recognize Hamas as a government, but also refuses to help Gaza hold an election. … And before you say “Israel shouldn’t be involved” then why are they in charge of pretty much every other facet of Gaza?

Because they are an adjacent hostile terrorist state, dedicated to their destruction, that they want to contain via leverage?