I'm not sure if I'm misunderstanding what you're suggesting, but my issue isn't that I want unlocking to be even more streamlined (since I'm fine with what I have and don't want fully auto unlock); it's just that I don't want the timeout that somehow started happening before it's even letting me manually unlock.
And yeah, I did try out various LLMs (including Claude) when it first started happening, but none of them were able to point me towards anything definitive about the cause, let alone the fix. In general I think it might be worth giving someone asking a question about something like this the benefit of the doubt about whether they've tried it, especially if you have outsized assumptions about how likely they are to solve problems. My experience is that when someone mentions having an issue, telling them "just ask Claude/Codex/etc." doesn't tend to end up being a very valuable contribution to discussions; people who are open to using LLMs for things will almost definitely have already done so, and people who haven't already are a lot more likely no to have done so for other reasons than the idea not having occurred to them. It's pretty hard to go more than even a few minutes into any sort of discussion about tech nowadays without someone bringing them up, so I think it's a pretty safe assumption that most people are well aware of them and considered them for whatever problem they're having.
To be clear, I'm not trying to berate you or anything but genuinely give feedback that might be helpful for you when trying to give advice on things like this in the future. Trying to give advice in a way that people will be receptive to can be difficult, especially when they're strangers on the internet, but I also think it's a a valuable way to spend time compared to a lot of other ways to spend time online, so I think it's worthwhile to invest time and effort into figuring out the best ways of doing it.