I also believe there's a strong way to fictionally justify attacking downed PC's so that you don't have to feel like the 'bad guy' for doing so. Just tell the players the enemies can perceive if you are dead or not, but simply don't know how long you are going to stay down if you are still living and so finishing you off before you have a chance to potentially get back up is typically their best course of action. Now they know they are going to be targeted and can plan accordingly.
It is an interesting thought on tactics, but rationalizations you give the players do not change what's happening. I think a more honest, "Guys, I'm having a hard time challenging you in fights, so I'm making this change..." is a better bet. Especially because the "issues" you cite are not universally considered problems.
You might want to consider using one of the death saves variants going around of the form, "After reaching 0 HP, the PC cannot take actions or cast spells, makes a death save every round, but can move 10'/round." I mean, the character is still moving, so yeah, it is clear they aren't dead! But the PC also gets the chance to get out of the way - and force the enemy to use movement to chase them down if they really want to do that.
3. Because no one wants to be downed due to risk of death being high when downed then whack-a-mole is completely eliminated - and midcombat healing before allies drop becomes a thing.
If the central issue is whack-a-mole, there's probably a better solution:
"If your character has not stabilized, healing stabilizes, but adds no hit points." It then requires at least two actions to apply healing - and actually gives a
reason to have medicine skill, come to think of it! Medicine to stabilize, then apply healing, but you've now used two rounds of actions.
There could be a special reason that you might hit on downed PC(depending on initiative order), but in addition to trying to eliminate still active target, you also force downed PC companions to spend non optimal actions.
Well, "optimal" is context-dependent. Having folks not die seems pretty optimal.