I appreciate it if my computer enemy is playing dirty against me, it keeps me on my toes and always looking for a payback. Room for improvement is still to be made, because other than falling back down, there is no penalty or consequence for having your teammates fall unconscious multiple times during a fight, and you can use that to absorb big hits for the rest of the team.
A suggestion would be to have teammates be counted as dead right away if they drop below a certain amount of negative hitpoints. Basing this treshold on the maximum HP of each individual character would be reasonable to me.
I think this behavior should be different from different creatures and how they evaluate threats. Animals are probably gonna go after the obvious threats. Zhentarim may stop to double-tap.
As for your suggestion: that's actually basically the rules, although 5E doesn't have negative hitpoints. It has to come all in one hit.
From the PHB: "Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death."
Additionally, it's the case that the "unconscious" condition turns any attacks from creatures within 5 feet into automatic crits. So that means that one attack on a downed character automatically changes the situation from three death saving throws to one all-or-nothing roll, and a second hit is certain death.