It is incredibly reassuring to hear somebody else find these symptoms - it took me weeks to narrow down the cause to the dodgy adapter, and then I ended up buying a $80 bluetooth adapter for my wired headphones, because I couldn't find an "ignore the stupid commands that the broken lightning adapter is sending" option...
What's the point of a headphone if you're listening to MP3/AAC → A2DP transcoded music?
Or if someone wants the headphone form factor, then 80 bucks is more than enough for an ok pair of Bluetooth headphones that won’t sound worse than the A2DP transcoded stream already is.
Also the cost differential that you seem to suggest exists between them is not apparent - I have seen examples of both form factors of bluetooth headphones as cheap as GBP 10.00 (or less) at a minimum, and you could spend way over GBP 500.00 on either kind if you so desired - you get what you pay for, though...!
I already have a pair of super-comfortable noise-cancelling headphones that I carry everywhere, and getting an adapter is more convenient than carrying wired headphones in one pocket and wireless headphones in another pocket :P