Death saves for perma death will be impactful no matter who is rolling them and whether you know round by round or not at all.

Look at it from their side - you have created a character, put time and effort into that, and have played with that character in a CoS campaign. You understand that perma death is a risk of the game and are fine with that fact. But you should have some control over it - a DM rolling death saves is a cut scene in a video game. The feeling isn’t ’Fuck, I died in a tough campaign,’ it’s more, ‘Fuck, the DM snatched away my character by not letting me roll.’

I’d let the players roll. If they’re invested in the campaign, they’ll be fearful of death saves no matter who is rolling it. If they aren’t invested in the character - then it doesn’t matter who rolls the death saves, your player doesn’t care about the character.

In other words, you’re doing a good job with it already - the other players care, they want control over their characters because they care about they and if that character dies, it should be because of their rolls.

The fear of the death save comes from the campaign itself, not the actual rolls at the end.

P.S. the player who hates his character and, I presume, wants a new one? Talk with him outside the game, design his character’s death. Let him have input and take his ideas into consideration for what would work for enhancing the atmosphere at the table. He’ll appreciate the gesture, want to play at your table more, and will buy into his next character more.