IDK if there are many of those things still being made, but sure, if there's any private info on the player and you're trying to be ultra careful, then get one without Bluetooth. I can also imagine (IDK if this has actually been done) someone using a bluetooth attack to turn the player into an audio bug from (say) the next room.

Players do tend to have built in microphones so you can use them as audio recorders. I had (and mostly still have) a bunch of different ones, though they were audio only and not video, or at any rate had tiny screens. There are tons of them on ebay.

If you're going to carry a big player with video and an ebook reader, why not just take your phone? Put it in a mode that blocks all calls. Or get another phone and use it with no sim and turn off the wifi. Then you can run your usual audio and video players. Dedicated players were once popular but now are very niche, since everyone uses their phone now.

FIIO makes some good ones if you still want to shop around.