tl;dr: Absolutely, positively, never do anything illegal for your boss. Don't do unethical things for your boss. Report breaches of personal information. Your employee handbook and policies, along with the law, are your guide. If you feel strongly about this, use an anonymous line/official reporting mechanism if it protects you from retribution. Don't trust HR, they're there for the company.
He asks me to put it directly into the manager-only forms and then he
signs his name to it. I presume there is no mention of my involvement
in this activity whatsoever. I suspect my peers have no idea I am
doing this. My own performance assessment (last year) read as if
someone else besides him had written it.
This is their decision to make, not yours. You definitely need to cover your ass though by confirming, in writing, that they want you to do this. If they refuse, you need to note the date/time of your request, and what was said. The access granted alone is unlikely to protect you from HR. They will protect the company, not you. They would probably spin it on you as a technique they've developed, and now you're on the hook for insubordination.
In his defense, this started when he had a ridiculous amount of direct
reports, but that is no longer the case. And, now that he has the
time, I'm not sure he has the interest to observe and evaluate the
performance of his direct reports in any meaningful way.
Totally irrelevant. Don't come to the defense of someone who puts you in a compromising position.
It is unethical to sign your name to someone else's work. But, there
is no structure in place to inform, without authority, the performance
appraisal process. It seems reasonable that an employee should know
It's not unethical at all unless there is a requirement that they, and they alone, do it. It can be considered part of their duties to have completed, not necessarily (albeit conventionally) by them. Most of these things are owned by the company, and not the person who wrote them.
It seems reasonable that someone who is more or less unobserved and
autonomous, and repeatedly consulted for this kind of input should be
at a higher level than myself.
Sure, but not really your problem.
I'm unsure how to face this crisis/opportunity. Suggest creation of a
structure for performance appraisal 'advice'? And insist my name is on
it? Simply cooperate and ask for the promotion? Look for another job
You don't want to have your name on it because without a title, it'll cause a row among your coworkers, and it will most likely be you who eats the shit pie. CYA, and carry on. Cooperate with HR if it ever comes to light, but avoid directly placing blame on your boss. You were merely doing your duty as their employee to the best of your ability.
You go to HR for things that are illegal, go against the business' code of conduct (or handbook) in a specific way, or would put the company's reputation at great risk. This article may help you.