Why is there no healing or recovery on the Manticore?
Often the discovery of new bases is few and far between.
Further exacerbated by the need to travel back to base for recovery so that you’re not exhausted on the next enemy encounter.
You can mitigate healing with med kits - but the loss of stamina and action points from pressing on without returning to base is a fairly big penalty.
I imagine that it’s purpose is very much to keep you around your home base. This and scanning seem to be replacemenets for the fuel mechanic which was in the earlier backer builds. It would make more sense if soldiers were easier to come by, encouraging dropping one party, and picking up another.
As it is now, it seems like some vital pieces of design are missing.
Because it is not a flying base? 
There is no facility where soldier can heal his wound or take a good rest. Unless Manticores are vip jets with medical cabin. 
As someone who has taken a nap as well as had a bullet wound patched up on a Super Stallion, I can tell you that it is entirely possible to do both in-flight.
You make it sound like humans are physically incapable of either sleeping or healing up unless they’re in a 5 star hotel. Trust me, when you need to rest, you can sleep anywhere.
And then, taking more in-game stuff into account, there’s also the fact that any damaged limbs and lost armor get replaced while on the Manticore. That right there is far more immersion breaking and unrealistic than recouping willpower mid-flight.
To further @endersblade’s point, Canada’s recent (and absurdly short, but that’s politics so nevermind that) contribution to the UN Peacekeeping mission in Mali was basically a modern MASH helicopter in the form of a CH-47 Chinook kitted out as a flying hospital with Combat Rescue soldiers stationed on it
However, this is not standard layout, so if we can get upgrades to the aircraft, I suspect that a medical bay upgrade would be one of the choices
As for the lack of recovering fatigue, that seems harder to explain. Unless it is combat fatigue, rather than just normal fatigue
I know it, but it has consequences. You can’t do it all the time as it has impact on your stress and overall mental health. This is not about taking nap. It is about releaving stress, about taking your ass out of tin box of death - well in PP aliens can’t shot down Manticore, but you probably know what I mean.
Then, why not. But not as a standard fitting of dropship. Phoenix Project probably was preparing to be able to fight, but not to join regular conflict with multitude of combat scenarios. They meant to prevent war or provide good solution to it, not to take part in it so having such stuff should not be typical for our ‘faction’.
personally, i think there should be an option to heal your squad while flying on Manticore, but spending medikit for each heal… MedBay should heal for free, while healing HP during travels would cost resources this way… after all, you can do exactly this first thing in combat, all you need to do is to take few extra medkits and sacrifice first turn which usually is not a big deal anyway…