It's just practical reasons.
Of course a taxi isn't more immune to crashing than another car, but taxi usage is by its nature occasional, so statistically, over your lifetime you're probably more likely to experience a crash in a private car than you are in a taxi.
Then all of the aforementioned things about the drivers being unqualified to fit seats, not knowing what size seat to use, storage problems, if it was rattling around in a boot it could get damaged.
It's not a perfect solution at all but it's necessary. If you're planning the journey in advance and have the means to, a parent can always provide their own car seat, and that works just fine.
The reason for 0-3 being able to travel without a seatbelt is that seatbelts are only designed to work on 3+ year olds (even most 3 year olds are far too small). Taxis are exempt from newer legislation - the older legislation said that 0-3 year olds must use a child restraint if available, if unavailable then they could travel unrestrained. It's meaningless to have a law saying you should strap a baby into a seatbelt - how would you even do it?? Safer to hold them, as that will stop them rolling and climbing around in the car. If you did crash they'd have no chance, but a seatbelt wouldn't protect them either.
I believe the parents who put the buggy car seat into the boot may have been breaking the law as if a suitable child restraint is available in a taxi, it must be used. But perhaps it was unsuitable - baby was too heavy for the weight limit, the car seat had been damaged, or something like that.