Why does the Atronach sign even allow you to regenerate magicka?
Because nobody would realistically use it otherwise. The magicka system in this game is very different than previous ES, as you can deplete your reserves with only a couple of big dualcast spells. Maybe if it gave 1000 magicka it would be worth using without any regeneration, but even then I doubt it

Wouldnt want to spend half the game fighting my flame atronarchs to regain mana like I did with ghosts in Morrowind. Gaming theory has evolved beyond this.
Although the Breton/Atronach combo is very powerful since you can be immune to magic once a day (laugh at dragon priests) and have a constant 62.5% reduction. Also, when fighting mages, they keep your magicka filled.
Much more than 62.5% actually given a Breton can get a passive 60% MR naturally (25 from race, 15 from Agent of Mara, 20 from Alteration perks) on top of the 50% from the Atronarch, which means a 80% reduction. If you go all the way in Alteration and get the perk for 30% Atronarch, then its a 60% on top of 80%, hence my previous 92% passive reduction figure. Lord stone only gives 88% and no magicka bonus, but no regen reduction. Sure the perks require an investment but Agent of Mara is a quest perk, so everyone should have it in due time.
As for the mana regen, I just tested it in game, and it only reduced my regen to 250% if my watch is any good.