> I would even expect if I quit browser with something written in the address bar (or search bar, etc), when it starts up again, it should be able to preserve this half-entered user input!
I'm pretty sure no one expects nor wants this. You're definitely in a minority here. The problem you describe isn't very technical, it's simply that you want something else than most people else and you don't want to take the time to configure your browser to how you want it to behave.
I think the lack of improvements in desktop UX in the last 10 years is making every desktop OS to slowly adopt mobile UX conventions. Some of those new conventions are a step back.
Personally, I don't like it when browsers implicitly keep tracking things when I close the application and then fire up some half-baked version of the same stuff when I come back later. If I want to remember where I am, I can easily bookmark some or all of my tabs.
Firefox has had a particular problem in recent versions where it seems to think it crashed on the previous shutdown, even though there was no user-visible evidence of this, and then tries to restore the previous session even when it otherwise wouldn't. That's not helpful if your previous session involved shopping for surprise Christmas presents for the SO who is now standing behind you as you open the browser again several hours later for some entirely unrelated purpose, in a totally hypothetical example.