Although I do name each and every member of my team
(though GF isn't exactly lacking in cart space nowadays...)
Although, I wonder how much space nicknaming takes?
In RB:
Name: 10 bytes
10 * 6 (for team)
+ 20 * 12 * 10 (20 PC boxes, 12 per box, 10 per name)
= 2460 bytes (about 2.4 kb)
That's not counting any extra bookkeeping required to manage those names, which would be in the ROM itself, rather than saved on the cart - I'm not sure how much space it would take overall on the save, since the RAM size was 16kB.
Maybe they saw that if they had nicknaming, that extra 7.2 kB (if they had 3 save files) would push them over budget on save space (45% of the total space devoted solely to nicknames would've been a lot, admittedly), so rather than cut down on the maximum amount of Pokemon you could catch, they decided to use one save file instead.