We need health regain mechanics for all 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.

Considering the amount of damage you take on Absolute or on Lethal, the suggestions I've made definitely would not make anyone invincible, not even close. They also in no way remove the teamwork. What they would on the other hand is making people not rely on very specific perks and builds just like Fenrir4life pointed out. He also mentioned the very reason why these mechanics are needed: "keep one bad moment from forcing players into Survival-horror mode for the rest of the run." Many times, it's not even in your hands to prevent this. Rng of the spawns are wild, enemies hit hard as a truck, the numbers are high, sometimes waves dont seem to stop. One bad moment, one unlucky moment and your health is gone. Also imagine playing perfectly and getting hit just before the enemy is cleared. Now you can not even get your contested health back and youre stuck with low health. I'm not against challenge, but harsh punishment upon rng of the spawns or one bad or one unlucky moment is not fun, nor fair. Many of us also did not pay for a souls like game where we need to avoid every single hit coming our ways, this is a horde action shooter game, and even though it turned out to be a bit different it is still a sequel to the first game. The first game was not easy, but you could gain health with executions.
About the ammo thing, depending on the enemies I face and depending on the Terminus spawn, I too find myself without ammo, even though I use the ammo variant of Multi-Melta as Heavy. I think a little bit of improvement might be good on ammo department but this is an "Idea" Ive opened for health regain, so that one is the subject here. I do agree that ammo thing can be quite problematic at the wrong parts of maps with Terminus enemy showing up as well as Terminus enemy showing up with other problematic enemies, especially for Heavy since it doesn't even have melee like it was stated.
And just for the record, I've sunk 400 hours into the game myself, in case you're wondering how much invested I am into the game.