I'm on a 5s though. I still have the headphones jack.
That doesn't mean your lightning connector is capable of putting out analog audio, it just means your headphone jack does.
AFAIK the lightning connector is fully digital and the D/A convertor is only in the adapter
Clearly not so if I've got a jack ...
if the DAC only exists between the logic board and the analog jack then there is a good chance it isn't being piped back in to the lightning port. Even then, I really doubt an entire pin on the lightning connector is dedicated to analog signal, so a DAC would have to exist on the other end of your lightning cable anyway; I suspect this is where your difference in quality come from.