Health does reduce damage taken, but I think with the numbers involved it generally is not a significant reduction, especially not when compared to armour (as I understand it, it can never actually reduce damage lower than the original roll - it can only counter damage bonuses from high accuracy or melee strength).
The tricky thing about LCS combat is that it's very all-or-nothing. You kind of need that guaranteed first round of attacks because if anyone with a gun survives with enough health left to counterattack, the odds are not terrible that they will fatally wound one of your squad members. Attrition really does not favour the player in site actions - they can only ever have 6 people, max, while the supply of heavily armed Conservative reinforcements is endless. So a hit against one of your people is orders of magnitude worse for you than a hit against them is for them.
Ultimately I think that no matter what you change in the combat, you're going to keep running into the issue of the fact that it's completely non-interactive. It's basically just an extended skill check, not really much different than picking a lock, except the consequences for failure are a lot more severe.
If you want to involve the Health stat a bit more, you could maybe have some kind of "Death's Door" mechanic similar to Darkest Dungeon, where when you drop to 0 HP instead of just dying instantly, you'd make Health checks each time you take damage, where passing the check means you still manage to hold on. This might make people with really high Health too hard to kill, though.