Zoom actually has in-browser WebRTC support, but they make such a concerted effort to hide this option. I can only assume they have some kind of incentive for wanting people to install a client on their machine. Tracking, analytics, metadata, who knows?

If when you join a call (without zoom installed) you click the download button, then let the zoom installer start downloading, then press the “I had problems installing” (or some phrasing to that effect) button, finally the join through the browser button appears.

Yes, you have to download the zoom installer executable every time you want to join a call.