Two part answer.
A) I prefer to listen to the output from speakers. It just sounds more 'real' from a large driver - possibly because of the amount of air moving, the harmonics and reflections from the room, the vibrations in your bones (if you can play that loud on occasion . . ). I always play the bass through speakers, either in the stack in the basement - rarely - or more often through the Hartke combo in the 'music room'.
B) As SWMBO is frequently hanging around the place and claims to have sensitive ears I do have to resort to headphones a lot of the time. I have a pair of wireless ones from Lidl. Cost â‚¬39.95 I think. They are rechargeable and I have never used the optional AA battery holder, relying on the built in battery. I tend to play for about half and hour to forty five minutes are a time before putting them back on the cradle to charge.
You're right, wired headphones are not convenient really, but I do use them when recording.