From a hiring manager perspective, I wouldn't say that it's game over, however. I've had plenty of devs get to the interview stage that have an experience vs skill mismatch, maybe 5 years instead of 10, but I consider them for maintenance work or our QA team that's less code heavy. For someone in this position, those roles might be a better fit or the candidate would have a place to prove themselves. Of course, you'd have to take a pay cut and take a mid-level or possibly a junior position. If I offered someone with 5 years of experience a junior position, it would probably be communicated that a bump could happen in 6 months to a mid-level role. A trial period of sorts.
The key is to be upfront. I've had devs try to bullshit their experience into something it obviously wasn't and it never works. I'd be blown away if a candidate came in, acknowledged their deficiencies, and said "I'm doing XYZ to figure it out and I really just need a chance to prove myself". Don't underestimate the importance of attitude and industry experience.
> What companies are most willing to give the "one year experience over many years" worker a fighting chance and get them up to speed?
Something about this question rubs me the wrong way. If you've fallen into the 1x10 pattern, no company is responsible for giving you a fighting chance. The burden is now on you. The advice I can give you is the same as any other dev, but probably more important, find a way to show me your skills and that you're working on improving. Open source work, code tests, side projects, anything and be able to discuss them.
The 1x10 devs that I already have are the ones who reject the opportunities given to them, who don't want extra responsibilities, and who aren't interested in learning new things. You need to give me confidence that that isn't you.