Atronach is not really a general mage choice. It's fine for the Alchemist-type mage, who is going to be exploiting Alchemy anyway.
It's possible to do a magic-user of any race, with any birthsign. The problem will be magicka management, unless you choose a high-intelligence race with some kind of magicka boost (either racial or birthsign), since Magicka is based on Intelligence.
There are other ways of boosting Magicka. You can wear items enchanted with Fortify Magicka or Fortify Intelligence. Also Fortify Willpower (assuming not the Atronach sign), because high Willpower makes Magicka regenerate faster. Also potions that do those things can be helpful.
Destruction is the big Magicka expense. Without a large magicka pool, you may run out of magicka before the fight is over. My successful Destruction Mages have been Bretons or Altmers with a magicka-boosting sign, and have still needed Fortify Magicka in some form.
If you use Conjuration more, and let a summon fight for you, your magicka will tend to regenerate before the summon spell runs out. Same is true for Illusion; spell effects tend to last long enough for magicka to rebuild. Conjuration and Illusion work well together; if you have enough magicka, you can cast "Summon Clannfear" (for instance) and then "Invisibility," and effectively disappear and watch the fun.