The randomness of the death saves themselves means that nobody ever says 'oh, he's got two death saves left, leave him for now'. In fact, the knowledge that fred is bleeding out badly while joan is hanging on in there helps the PCs decide who to attempt to save first, which is all to the good.
I suspect the point is that you
do have someone say, "Well, Jed has been down for a few rounds but only failed one save, so we can keep focusing fire on the Kobold Sorcelator, and tend to Jed next round." I've definitely seen people say, "He's got two death saves left, we can wait to get to him."
If the DM rolls, you know that you've got maybe one round of safety... but after that first save or two, you have no idea if you've got a few more rounds to tend to your companion, or if they are already at the verge of death.
Now, you can still make a tactical decision here - do you make sure your friend survives, or focus on taking down the bad guy? But your decision now depends on unknown variables. That can heighten the tension - you now are taking a chance with a PC's life when you make your call, rather than reducing it to an easy equation (If PC is within 1 failed save of dying, tend to PC; else, kill monsters.)
Now, that certainly isn't universally true, and even when it is, there can be advantages, as you say, to making those decisions while fully informed.
All that said, I'm likely to stick with players rolling in my own games, if only for those occasional moments of glory when players roll 20s - and the fact that it lets players feel like they have some agency over their own fates, even if the dice roll is just as random regardless of who rolls it. But I think there is room for campaigns that want to keep things more immersive, or for systems that bridge the line between the two.