While it’s true that players turn off charging when the battery is charged, they turn it back on again when it goes back down a little so it’s getting charges a lot more than it should. That tends to shorten their life somewhat.
I know with laptop batteries leaving the battery in the laptop all the time when it’s always plugged in shortens the life of the battery enough that most people will replace it if they need to use it in a year or two. My experience with laptops has been that the batteries last 3 to 5 years if I don’t let that happen and a year if I do. Of course the laptop is still usable and the battery doesn’t die totally so as long as it stays plugged in it doesn’t matter much. But I generally remove the battery while it’s plugged in.
I haven’t left an MP3 player plugged in that long but they’re the same type of battery (Li-ion) so presumably the same thing will be true. But if you plan to always leave it plugged in it might not matter what happens to the battery.