Delicious player does in fact work and allow you to serve up more than 15 files on the same page. I knew this already because that's the one Jonathan Coulton's web site uses, and his site works.
I didn't, however, know how to implement the Delicious player until you sent me that link. That's a good one and the actual coding required on my side is incredibly simple. Thanks for that link.
One of the issues, though, is that it doesn't meet my specific needs. It's one of those javascript/CSS players that I don't want to use, because, when you hit a button, the page shape and layout has to change because of the extra text that appears next to the button. There might be a way to work around that by copying and editing their Javascript code, which might be interesting to try some time.
I wonder if that Javascript trickery is how it's able to do its magic? Perhaps it's not loading one flash object for each song, maybe it's a single flash object active at any one time, and the javascript trickery is what feeds each song to the single flash object. That would definitely work around the bug since the bug is only if you have more than 15 or so flash objects up at a time.