We need health regain mechanics for the classes

I don't agree with that.

There are already enough ways to regain life. 3 classes out of 6 offer this possibility.

 - The Bucward with the banner.

 - The Vanguard with the 20% of heal on extremis.

 - The Heavy by raising a teammate full life + armor.

 - The Stims 

- The relic 

 Giving healing mechanics to everyone will just fill the game with solo rambo and kill the little teamplay we can have. People will feel invincible and will just wander around the map looking for majoris to kill to regain life. 

 My brother protects me and I protect my brothers!

The extreme scarcity of healing is responsible for the undesired player behavior of "basically always use this same handful of perks that grant healing."

In response to this behavior, they have nerfed vanguard self-healing, planned to nerf Bulwark party healing (but did not actually implement it) - neither of those things are necessary.

It's the same thing as fencing weapons: they created a game environment that has certain right answers.  Rather than try to nerf those right answers to increase build diversity, they should acknowledge that those answers are in direct response to the game environment - and make some version of them baseline to free up design space.

My personal favorite answer is "Gain 10% of your missing health on execute, OR any contested health, whichever is greater."  It maintains the sense of scarcity that they do want, in that you really can't use it to heal to full - but it will help with stim-hoarding and keep one bad moment from forcing players into Survival-horror mode for the rest of the run.  Vanguard perk granting an additional 5% of max health on top of that would still have a playstyle impact, but would no longer feel "mandatory" as there would be another reliable source of sustain.