It would affect the atmosphere for sure, but perhaps not for the better.
There's a reason tactical combat games typically don't depict enemies begging for their lives or screaming in agony or calling out for their mothers when mortally wounded, even though these things may be realistic. You don't want players to feel sympathy or pity for the enemy when you're expected to slaughter hundreds of them over the course of the game. Seeing enemies desperately try to save the lives of their wounded comrades, and being incentivized to gun them down in the process, is probably not the gameplay a lot of players are looking for.
There's also the fact that, in Jagged Alliance, first aid can save a merc's life but usually doesn't get them back in the fight, due to the penalties they suffer even after being bandaged. This is worth in the long run for the player, but since you're not expected to leave any survivors among the enemies, this means first aid on the enemy's part is just a waste of time.
Your stories sound a little strange to me. In the first XCOM, you absolutely could pick up unconscious soldiers. Their bodies were considered objects an appeared on the inventory screen. In Jagged Alliance 2, no one remains unconscious after being bandaged up, unless by "unconscious" you mean Nails's energy was completely depleted, and even then I wouldn't expect it to remain so for several turns.
I am 100% against this idea.
If a mercenary is badly wounded in the course of gameplay and has to be carried to safety? Fine, that's great. If a quest involves an NPC being wounded, or unconscious, or missing a leg, or whatever, and has to be carried to safety? Fine, also great. Both of these cases occurred in X-COM 2 (the new one) and were fine.
What I don't want is to load into a map and be told "oh no, Mouse tripped over a rock and is now dying, you better carry her to this waypoint in five turns or she'll bleed to death!" That's bullshit and not what Jagged Alliance is about.