I carefully check all the links periodically, and at no point have I found anything "wrong" other than some of them becoming 404s over the years, at which point I replaced them with something else or removed them. Except one I removed about 10 months ago - the page looked fine, but on closer examination, the site overall looked like it was aggressively targeting a particular keyphrase. Everything currently on the page looks good - it's either from an established, respected site or a little blog that is not doing anything nefarious as far as I can tell.
I'm going to consider your advice, but I just discovered something else with this one. Last year, I changed the domain name for branding purposes, so many of this page's backlinks were redirected from the old site via 301s. Then I also changed the URL of the page, hoping for a completely fresh start. But at some point, the original URL (with the old domain) has started redirecting to the new URL for the page. This has to be a Wordpress thing, and it certainly wasn't happening when I first changed the URL, but it must be an update thing (don't ask - it's a very peculiar thing, but I can see how it's happening). So my original backlinks have been going through not one but two redirects, and I think it's possible the SE's would all react badly to that. So I've changed the URL back to its original one and it's now only going through one redirect. If the engines respond in a few weeks, cool, and if they continue to hate the page, I'll take your suggestions.
I really can't figure out anything ON the page for them to hate so much, and neither has anyone else who looked at it. It has to be an off-page issue, I think, so we'll see.