How can I upgrade my "obsolete" Pixel to a current/secure version of Android? I have a first-gen Google Pixel.  It came with Android 9 (Oreo), then after I'd had it a while, upgraded itself to 10 (Pie).  Just about the time Android 11 (Q?) came out, I was notified that Google would no longer supply updates (not even security updates, it seems) for my phone -- then not yet quite two years old.
Sure, most people buy a new phone as soon as the purchase agreement runs out on their old one.  I, however, bought the Pixel originally with the idea that, if I was going to spend $750 for a phone, I'd make it as future resistant as possible.  I bought the best camera, fastest processor, and largest storage (128 GB) that I could find, which is how I wound up with a Pixel (Apple wasn't in contention, because Apple).
Now, however, I have to choose between buying a new Pixel 4a (for less than half what I paid for my original Pixel), essentially the same phone I have but with very slightly upgraded specs (camera hardware supports longer exposures, CPU is slightly faster), in order to get two years of updates on a new Android version, buying a Pixel 5 or something else to get a significant hardware upgrade (and paying as much as I did originally) -- or continuing to carry and use my original Pixel, which still works fine, holds charge well, and is nowhere even vaguely close to full.
The only reason I have to think about this is that I'm apparently no longer eligible for security updates, meaning there's the possibility that a flaw that's been fixed for years in newer versions might let someone take control of my phone, steal my photos and storeds passwords, take over my email, etc.
Is there a way to upgrade manually, legally, with a non-rooted phone, and have the upgraded system work correctly with my hardware (camera etc.)?
The phone is on Verizon, if that matters (not sure why it would, but not sure it wouldn't).
You could install LineageOS for Pixel 1 (sailfish).
However as Google does not provide updated kernels (and drivers) the updates will only affect problems in the Android system itself.
The disadvantage is that the whole system is no longer genuine regarding Android SafetyNet check so that certain apps (especially payment/banking apps) may refuse to work.