sigh. First of all i said this way or another. Maybe using self heal was a bad wording, but the point stays.
Zen has 150hp in form of shields that regenerates itself by 30hp per second if he is not taking damage for 3 seconds - it still gives him 75% of his hp pool back without depending on others.
Lucio heals himself for as long as his song is set to healing, no cooldowns (except amp it up but it just makes it faster but not only for his self healing but also for anyone inside the circle so it’s a win win for everyone). Gets 12-ish hp back every second.
Moria heals through damage, there is no cooldown on her secondary fire, she can also simultaneously deal damage and heal the team if she use her orb. She gets back 30hp per second. She can also sacrifice her orb and heal herself by it making her the second healer that can apply her own kit to heal herself, but comparing to Ana she has also other way. Cooldown is 10s on orb BUT like i said it’s not her only way to self heal.
Brigitte similarly to Moira get self heals through taking damage but differently to her she MUST deal damage to heal others too (i know she also has repair kit but it’s on 8s cooldown). 16hp per second for the duration of 5 second. Inspire is apparently on a 1.5 cooldown but it’s not that big of a deal.
Mercy has regeneration just because. 20hp per second after not taking damage for a single second i think. She doesn’t heal herself with her staff, doesn’t have shields that would regenerate. she just automatically regenerates her hp because reasons.
And then we have Ana who has no mobility, no regenerative shields, no armors (blah blah blah, brigitte doesn’t need them either), no self healing “just because”, no self healing through damage. She is the only hero that has to either apply some of her kit on herself (with a really long cooldown) or being dependent on the other healer/health packs.
So yeah zen has something than can be called self heal ability.
EDIT: friendly reminder that Mei, Reaper, Bastion, Soldier and Roadhog have some sorts of self healing ability. There is also Sombra who’s hard to put into these categories as all she does is hacking medpacks to respawn them faster but since there is no ult charge from that there is no point for the team to use them (it only slows down healers ult build) if it’s not necessary so it can be more or less treated as sombra’s way for “self heal” or at least to be less dependent on healers.
That being said i absolutely see no reason why giving more self sustain to ana is that bad of an idea