I think you mentioned this before. If a player did this in my game, they would not be invited back. It is insanely disrespectful IMO.
While this sometimes happens, more often than not what seems like dumb luck (a critical hit, for example) only led to death because the player insisted on playing their PC as though they were invincible.
For example, in my last session a week ago, a player has the chance to have his PC disengage and reform with the rest of the party, but he insisted on attacking even though he was already injured and below half HP. He hit, but the monster survived and critted him, for instant death.
So, while some people might say, "Oh, that isn't fair, it was just bad luck." My answer: "Maybe, but what happened before that?" Odds are, the PC put themselves at risk and should have been considering other options.
That being said, sometimes it is just "dumb luck", but IME more often than not there are underlying circumstances.
Yes, that is his job, but not to the point of suicide. If he is pressed, he has to be able to fall back and have others pick up the slack. Unless you have a small party, most groups have at least two front-liners IME, and they can trade off duty if the encounter permits.
It isn't mitigated in that sense, it is withstood. Those PCs also tend to have the most hit points purposefully for taking that damage.
My first suggestion would be a coma until they finish a long rest. You mentioned not doing time-sensitive missions, so that might work. If a long rest is too long, maybe just a short rest?
Another option is allowing them to spend HD to stay up, using their reaction maybe. But while at 0 hit points perhaps some penalty, like half speed, disadvantage on attacks, etc. How severe you want this is really up to your group and what they feel good with.