Sony was fond of resonant supplies as well, using current feedback to solve base drive, via a saturable reactor to control oscillation frequency and thus power into the resonant tank. Trinitrons for example often used this.
The common CFL does the same thing, without control (or if it's controlled, it's by fixed saturation in the drive transformer).
1) switching them fast and hard is very, very difficult if not impossible
To avoid excessive on state losses the transistor must be turned on very hard, unfortunately this leads to extremely high carrier concentrations that must be extracted from the base before the transistor can turn off, all techniques used in small signal applications to enhance switching behaviour can’t be applied (baker clamps of various topologies and high gate resistances) as they would result in massive on losses
What's fast? For the time, fast was 10s of kHz, then 100s. MOSFETs took over at exactly the same frequencies; I wouldn't at all say it was difficult!
Base drive isn't hard to arrange, but it's really only practical through transformers, which I suspect is one of the main reasons they've fallen out of style. Transformers are componentized labor, and labor costs are continuing to rise.
You don't avoid storage time, but doping profiles are optimized for rapid fall time once t_s has passed. Typical late generation 800/1.5kV line output transistor did 2us storage and 150ns rise, not a bad ratio at all!
Alternately, you can use Baker clamp style drive methods, but they're really only practical with bootstrap drivers (yes, they've been used with BJTs as well!), especially in integrated circuits (where the dozens of additional transistors are practically no cost, and very complex drive methods can be arranged, perfectly tuning Vce(on), hFE(on), rise and fall).
A number of LT switching regulators are actually such things! You can tell from the huge voltage range (say 3-40V operating), low required bootstrap voltage, and the prominently placed NPN in the block diagram.
And also, higher voltage drops aren't much of a problem at higher supply voltages, for example an old fashioned SimpleSwitcher (or, lord forbid, even a MC34063 and its ilk!), with its Darlington output, isn't much of a problem with a 24V supply where the voltage drop only amounts to a maximum say 90% efficiency. Boo hoo!
No, certainly not going to compete with a modern synchronous regulator pushing 90-95% [overall, not just conduction], but they were competing with linear regulators of 50-70%, an easy win at the time.
2) the NTC characteristic of BJTs makes parallel configurations very prone to thermal runaway, and thus dangerous
3)current gain of power BJT’s is very low (in the order of 10/20), requiring very high drive currents/powers (this is very annoying especially for the high side where every transistor (two in a full bridge, three in a VSI) needs its own isolated DC/DC converter
Of course, they always made BJTs in whatever size you needed, so this was never a problem. To this day, you can get another bipolar device, the SCR, in entire-wafer models. No worries about matching there.
The biggest BJTs I've personally handled, were just some triple-Darlington style inverter bricks, good for around 10kW I think it was. Was a kinda-oddball motor drive, IIRC its contemporaries were all SCR and it was the one model that wasn't?
Tim