You could also use the existing MP3 player with a bluetooth transmitter dongle. I have successfully used the following bluetooth dongles, but, plugged into an airplane armrest instead of an MP3 player:
https://www.amazon.com/gp/product/B082G2Z1WShttps://www.amazon.com/Twelve-South-Transmitter-Headphones-Airplanes/dp/B0BB8FVKFCThere are many more like these. I do not specifically recommend any specific brand over any other. They are plentiful and varied; perhaps searching around for things in this category, you can find one in a form factor that meets your needs, based on their features. For instance, the "Airfly" one that I linked above has an alternative "duo" version available which lets you pair two sets of bluetooth earphones at a time, for couples listening.
The only drawback is that you must recharge both the dongle and the MP3 player. If you've ever had the MP3 player run out on you at a bad time, then you're basically doubling your chances that that'll happen to you.
If you don't want to use a separate dongle, then, as you already mentioned, almost any smartphone (even the cheapest ones, even if you don't buy a cellular plan for it) will work as a perfectly serviceable MP3 player with bluetooth built in. There are multiple available audiobook apps for them which you could experiment with.
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