Does my phone need to be on/connected to the internet for it to work? That's always been the weird thing for me about Whatsapp, is that your phone seems to be the "server" for the app.

Since a few weeks/months ago, no. Before that yes.

Except now it won't sync older messages and sometimes messages only get delivered to your phone but not the desktop. They took a decade to roll out a desktop app and even then it's a mess. WhatsApp has been degrading under Meta's ownership.

They fixed that fairly recently.

The old website/app was very primitive. Only this year did multi device linking come out of beta.