Soraka is the only champion that has done health-costs really well.

Soraka's health-costs actually makes you play around it in a meaningful way. Zac and Olaf does it okay to some extent, but not nearly as well as Soraka.

What makes it work so well is that, if she so choose, she is free to eat through her health rapidly by spamming her heal, which makes her maximum output put her at much higher risk. This is combined with her source of regen, which is a skillshot. Her sustain is therefore by no means guaranteed, which allow some interesting choices for her between how much health she decides to spend depending on how quickly she think she can manage to regain her health.

Regen from auto-attacks are problematic because it's too automatic, literally. This makes it a stat-check and low risk, which doesn't really change your play, high health or low. It's just a snowball-mechanic at this point, rather than something that defines a champions playstyle. Therefore, on someone like Aatrox it will only add a speck of nuance, at best.

Similar issues are found with "sustain-tanks" without health costs. An Akali that sustains through retaliating damage doesn't play around it per se, it just puts her even further ahead the more fed she is. Same goes for Warwick.

Mundo is also really bad in this regard, as he doesn't even have to auto-attack. He's effectively just a ball of tank rather than someone trying to balance his loss- and regeneration of health. He's not a regen tank, because the exact same result would be achieved with just a lot of health, especially because of how innately tanky he is. It doesn't help that his health-costs are so small.

Bottom line, the cost must be meaningful and much higher than the regenerating effect if not managed really well. Auto-attack regen or flat out % of damage are bad mechanics for a sustain-tank because it's too predictable regen.

I hope those champions that are supposed to be about health-costs are reworked with Soraka's design as a guide.