Exhaustion mechanic can be shoe-horned into this, but I think it might best be reserved for, well, exhaustion. Like going days and days without a break in a harsh environment.
For a life-draining type effect, you want something that is more dangerous to fragile PCs than to front-line PCs, or you run the risk of punishing front-line PCs, while it being a trivial problem for "fragile" characters.
And I get it, it should be scary for front-line PCs. But I'm saying per unit it should be even scarier for back-line PCs.
That way when the undead ambushes the back line, it isn't "yawn", it is "omg run away", while the front line remains credibly able to fight them and keep them off the back line.
And to be explicit, this is an advantage that both Strength Drain and max HP drain have over exhaustion; front line PCs have higher strength and more max HP typically.
In addition, your change has a side effect of making Strength matter even less than it does.
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So, I might suggest combining your two mechanics -- death saves and life drain -- into one. And making a mechanic that merges them.
Every death save failure can cause the same effect as an undead "vitality drain". And we design the mechanic to care about both HP and Strength.
One option is to give you (Max HP/10 + Strength Bonus) vitality drain before you die.
Enervation: When you fail a death saving throw you gain a point of Enervation.
If you have at least 1 point of Enervation, you suffer from 1 additional level of Exhaustion (if you have no Exhaustion, it acts as if you have 1, etc). This can kill you.
If you have more Enervation than you have Strength bonus, for each extra point your max HP is reduced by 10. If you have a strength penalty, the first point of Enervation reduces your max HP by an additional 10 for every point of penalty. (Ie, max HP reduced by 10*(enervation-strength mod) if enervation> 0, and never increased).
At the end of a long rest, you recover 1d4 plus your Strength modifier (min 0) levels of enervation. Lesser Restoration recovers 1 level of enervation, and Greater Restoration acts like a long rest.