A single one would require a ported app to understand that it worked with bigger registers in order to take advantage of the bigger address space. Some really old computers did work like that, but to allow several apps to coexist in memory, not to give each app more addressable memory.

The segment register solution does both almost for free in most cases. Code gets a segment, data gets a segment, stack gets a segment... and you are free to use all of memory if you bother to use the ES register. In many cases, that could be isolated to a small piece of the program so that was also quite cheap.

As someone who wrote code in assembler, Pascal, and C for those machines: it wasn't at all a nightmare. It was quite easy and simple.