why does it sometimes remember I’m logged in, after deciding I’m not

I had that problem when Lemmy was under constant DDoS attacks, almost a year ago.

iPhone 15 Pro is too slow to run this web site reliably

You have both upvotes and downvotes so I will assume you are not the only one with these problems. In my experience Reddit website either glitches itself or glitches Safari every now and then.

Why does it sometimes get stuck trying to draw the Home Screen

Sounds like iOS issue, not Lemmy.