I know one does occasionally run into a mp3 that won't play on iTunes for no discernable reason, and maybe this is the reverse. There are media information tools (I use an older one called MediaInfoMac from when it was free) but I don't think I have seen anything that told me why it wouldn't play in iTunes. There is a mp3 repair tool called MP3Scan + Repair. Usually the problem is, as I said, the other way around with iTunes not being able to play the mp3.
One complicating factor is you say one file will play. It still isn't clear to me if it is just the first file that plays regardless of the actual file it happens to be, or it is a specific track and none of the others will play regardless of position. If it is one specific file that indicates a problem with the encoding of the others. If it is just the first one the player happens to play that suggests something is crashing the player after that first track.
You may just have to figure out some way to get this player to work if you want to continue to use it. The player is playing one of the iTunes tracks without issue so it isn't a basic iTunes encoding issue. How old is this player? Many, many years ago they changed the style of tags used to add information to mp3 files and perhaps this player can't use the newer style?