Single-ship ownership worked in the previous games because the gameplay was built around it.
Elite 1: the Cobra III you get can be outfitted to do both good trading and extremely powerful combat performance at once. There was no real outfitting limit - you could carry all purchasable equipment simultaneously on the Cobra III.
FE2/FFE: the ships are basically only distinguished on speed, hardpoints and capacity. So once you've got to the ship with the right balance of speed and capacity for you, there's no reason to change. The popular ships from this era - the Panther, the Imp Courier, the Asp - all sat at particular sweet spots of speed vs capacity. While the smaller faster ships did have some limitations on capacity, you could still carry "one of most things" on the Asp, definitely one of everything on the bigger ships, and the occasional bit of reoutfitting was straightforward.
Elite Dangerous: a medium multi-role (Krait II or Python) can still be built to do all the Elite/FE2/FFE professions and a little bit more besides - the Krait II which I fly almost all the time can do trade, combat, missions, exploration, surface operations, and light recon/research duties without refitting. Despite being able to own multiple ships, I rarely actually fly the others...
However Elite Dangerous adds mining, AX combat, and CZ combat as activities which are tough-to-impossible to do with a multirole build, and really benefit from a specialised ship. (Yes, the previous games technically had mining, but it was so terrible it was best just skipped)
Elite Dangerous also has the large encouragements to make use of ship storage that:
- unlike the previous games, where Jameson-like "sells everything" stations were relatively common and straightforward to find on the map, Elite Dangerous outfitting is essentially designed to be incomprehensible. If you sell a useful module it might take quite a bit of hunting to find a new one.
- as pointed out above, engineering also encourages hanging on to modules
- several ship hulls have been optimised for high-end combat performance at the expense of most other capabilities, in a way that wouldn't be possible under the outfitting models of previous games. Every ship in the previous games was essentially a multirole (and had to be, of course)
So ... would Elite Dangerous have been a better game if it had been designed, like the others, around finding the ship you wanted and then sticking with it, all ships being largely capable of doing anything? I think probably yes, for me - the minor inconvenience of having to go and fetch a different ship (a whole two systems away!) means that if I can't do it in my Krait I'm generally not interested - but it would mean a more creative approach to outfitting to allow things like mining to work.