I checked out one of the files, and the first thing I saw was that the Song Start was at 0 1 1 1. That's always a bad sign. Even though Logic lets you position the start of a song before bar 1, it's never a good idea to use that feature. Doing that (whether intentionally or by accident) is a primary source of song corruption and/or generally wonky behavior, and might (just might) account for the problems you've been having.
Suggest you move all of the regions to bar 1 1 1 1, then try and move the song start marker to 1 1 1 1 as well. I tired that here and the song start moved to 1 1 1 1 on my second attempt (it's finicky).
That aside, the song played fine here immediately, but that's not to say that you'd have the same luck on your system. Every system is different, even down to the way preferences are set up. So... Suggest you take my advice above and see if you can get your songs to play.