It really depends on how we left things.
These days we are more likely to reschedule if someone can't make it, but if rescheduling in a timely manner becomes an issue because one or more other folks who could make the scheduled game have other plans that can't be worked around, we don't think it is fair to push further back or for someone else to potentially miss it, and play anyway.
What happens to the missing player's character - as I mentioned above - depends on where we left the game. If, for example, the party is in town, then the PC might be sleeping off a hangover, suffering from "bog flu," or out doing research for a broader campaign plot (and when they return I try to give them some event or info nugget to bring back to the others).
If we were in the middle of an adventure far from civilization or in the middle of a fight, I usually run the PC in a kind of minimal way, and have a designated player do the rolling and keep with that approach until there is a reasonable narrative opportunity for the character to split off temporarily. But while they are present, they are present - cast spells, fight, defend others, and generally act in a way that we all agree fits with the character's approach to things.
When we were gathering for the beginning of the conclusion of our Ghosts of Saltmarsh+ campaign, the druid's player did not show up and we soon discovered that an unintentional double-booking meant he was out of town not even realizing he was missing D&D until that very moment, I just ruled that he was sleeping off celebratory night with the locals - "we're probably going off to our death tomorrow!" - wildshaped into a rat-form in the hood of one of the other PCs. This way, when he did return the next session, he could just wake up and be as deep in the sahuagin lair as anyone else - but I could also use his character as a kind of potential deus ex machina, if need be.
I personally would never be satisfied either as a player or a DM with a character being present but not actually being "present" in a narrative way that makes some kind of sense. Does this mean that a PC could die when their player is not present? I guess so, yes. Though this only happened once (back in 1993 or '94) and in that case, hers was one of 3 PCs who died, and she ended up making a replacement character she'd play for years longer and ended up liking more than the dead one anyway - so it worked out without hard feelings - and I would make the risk clear.
I guess game expectations also matter. If you are playing a grindier game like Shadowdark or DCC, a dead PC is no big deal. Or a high level game with access to resurrection magic, same deal.
We do try our best to work around a potential absence - for example, we've had players Zoom into the session from home - when they were feeling up to playing but were still potentially contagious for example.
We might also completely call a game if the reason for a person missing is so serious that playing just feels wrong. It really depends on vibe.