We've noticed this behavior on iPhones from time to time. Typically when you choose to stay logged in, we store at least four cookies with varying lifespans in an attempt to work around various browser quirks. However, looking at the logs, your browser was dropping all cookies, regardless of lifespan.
Sometimes, iPhones do this on purpose to conserve resources, especially if you're running out of both memory (RAM) and storage space. Both are more likely to happen if your phone is trying to update, since updating is relatively resource-intensive.
There also seems to be a bug that causes it to trigger intermittently until the device is restarted. This is very rare, and I've never been able to reproduce it myself, but it's happened a handful of times. Given that you're also experiencing slowness, I'd definitely try a full restart.
As far as changes to our site go, we didn't deploy anything new over the weekend; our last update was Friday, and it shouldn't have affected sessions. (It's possible it caused some people to be logged out once, but it wouldn't have been a repeating occurrence.) We tested everything on iPhones before pushing anything live, and we didn't observe any performance issues. Most of the changes weren't on the frontend anyway.
We have not observed increased login rates.