Hi fonograph
Kleinstein has the answer.
Thought you may like to know that, with similar thoughts to you, I once made a current audio power amplifier to see what it would sound like driving a speaker.
With small LowFi speakers it made little difference: they sounded as bad as ever. But on decent speakers the sound seemed to vary according to the type of enclosure.
With an infinite baffle the sound was bass heavy but not unpleasant and the top seemed to be unchanged.
Transmission line cabinets were more base heavy and base reflex speakers were even more base heavy.
The general sound, apart from the base enhancement, was a clean effortless sound that was somehow unnatural.
Hope that makes a bit of sense.

At one time I think Phillips produced some speakers that had built-in current amps, but they were part of a servo loop that used a feedback transducer attached to the speaker cone. The servo speakers didn't catch on though.