If you think about dying in terms of rounds and what that means for time, a death saving roll happens every 6 seconds. This means that a character knocked out has a minimum of 12 seconds (one roll of 1 for the double fail, and then a roll below 10) and a maximum of 30 seconds (2 successful rolls and 3 failed rolls) before they die (assuming no additional attacks and such).
Because this happens pretty quickly, I would keep turn order going after the combat is resolved. Your party might only have 6 or so seconds to get to each player and stabilize them. This can pose a challenge for the long ranged magic user to make it to the front to stabilize the fallen fighter, or vice versa. It's always possible they fail the stabilization check too, making for those 6-30 seconds to be pretty intense.
I find a bonus is that this additional last chance to die prevents some players from using kamikaze tactics where they think "who cares if I get knocked out? I'll still be stabilized right away." Not necessarily in my games.
For me, there are a few points to consider, both out-of-character and in-game. I will split the answer in these two sections.
Out-of-Character points
First, let us consider the players and the DM, rather than the characters. The point here is that sometimes, the feelings of the people playing are more important than the in-game consistency (unless, obviously, the people playing want consistency) and you can make your decisions based on that. Note: you are one of the people playing as well, so your feelings matter too.
What kind of DM are you?
There are "Killer DMs" that want to kill your players. Also there are "kind" DMs that will pull lots of punches and even create some deus ex for the party to not be TPK'd in scenarios the characters totally should die. If you already have a favorite style, follow it. From your (now removed by edit) comments on yourself, you probably don't, so I'll keep it as if you are a versatile DM that can go along with either option.
How do your players feel about PC death?
This is something common to talk about on Session 0. Will your players feel bad if their carefully-created-with-a-lot-of-work-on-background PCs die? Will they accept it and understand that probably what caused them to die was a bad decision by them, not you being an unfair DM that wants to just kill them and did it throwing a too hard encounter at them?
Remember, the main job of the DM (and the players, but we're talking about DM'ing) is to make everyone have fun. While this might be done by being strictly fair and rational and playing it completely in-character, it also can be done by metagaming and just deciding the Red Dragon was in a happy day and didn't want to kill the party, because this is what makes everyone happier and being able to keep having fun.
About myself:
While I'm not a Killer DM, I usually warn my players that if their PCs do something too dumb (entering the dragon's lair alone while making alot of noise) their characters will die. Usually my players are fine with that, and I even pull punches sometimes, mainly if they are getting killed because I'm hitting too many 20s in the d20 and too many 6s on the d6 damage, not because they were dumb. I'm guilty of some TPKs but most of them are on the players. Except for some edge cases, I never had much trouble with players feeling wronged by deaths.
In-game points
Then again, many times we are interested in keeping the in-game consistency and being "fair" - meaning the hostile characters will do what makes sense them to do.
Why is the combat happening?
First I'll recommend an interesting reading on combats here. It's talking about dnd-4e, but it applies to 5e as well. Essentially, the hostile characters have a reason to be fighting against the party - there is some conflict we are trying to solve (through violence).
Based on the motivation behind the combat, you usually can sort out what makes sense. Bandits/thieves want to steal the characters, not kill them. A Dragon protecting his lair might just want the characters to leave, dead or alive. Wolves can be really starving and eat the characters, finishing them off.
Reminder: Although you mention "when the monster description or adventure don't dictate what happens", sometimes they do.
For example, the Goblin Ambush in Lost Mine of Phandelver explicitly states
In the unlikely event that the goblins defeat the adventurers, they leave them unconscious, loot them and the wagon, then head to the Cragmaw hideout.
Considerations on the "Usual"
From my experience, TPKs are usually seem as a bad thing, both by players and DMs, since they mean the game essentially ends, it's a Game Over and you have to restart with different characters with different personalities and motivations and recreate everything. Mostly we will just let the party live and punish them some other way (like looting their stuff). Note that this might be biased.
Also, usually, the hostile characters motivations aren't simply "Kill the party". Check, for example, Strahd on Curse of Strahd.
He wants to mess up the party, make them betray each other and maybe even kill each other, but he himself doesn't kill PCs, except when fighting for his life in his lair. Note that he is the personification of a villain that's pure evil, and even he doesn't kill PCs for nothing.
Usually the party is either being an inconvenience for the BBEG plans, invading a lair or something on these lines. In neither situation killing the party is needed - scaring them, throwing them away or something like this is enough.
So, from my experience, the answer to the dnd-4e question still applies - Most of the time, either for OoC reasons or for in-game reasons, the party won't be finished off, but punished somehow else.
Best Answer
You could roll initiative
One way of resolving this is to roll initiative. This at least resembles how it would be handled in (or at the end of) combat. However, as you say, it does seem a bit excessive...
Or act out of initiative
If one of the party has spare the dying or a healing spell, or a healing potion, then this becomes trivial, since they cast the spell and then no-one is dying. Even if the dying PC rolled a 1 on their death saving throw, they're back up (or at least stable) thanks to one casting of a spell.
However, if no-one has those spells, then it's down to death saving throws and medicine checks. This therefore becomes a skill challenge rather than a combat scenario. This does not need to involve the entire party, so initiative becomes irrelevant.
Rather than having everyone dogpile the fallen PC with Medicine checks (see this question about skill checks and why it's not a great idea to have everyone repeatedly try the same skill check), it makes sense if the PC with the highest Medicine check (perhaps helped by another PC via the Help action, which gives advantage on the check) trying to stabilise the fallen PC as the PC is making death saving throws, alternating between them.
@Yakk has pointed out in comments that this makes stabilising a dying PC harder than it would if everyone were allowed a try, as is commonly ruled during combat, but I would argue that, from a narrative perspective, this doesn't make much sense. Given that rounds of combat are 6 second, stabilising someone within 6 seconds is already cutting it fine, but if each member of a party of 4 trying to revive their dying party member, what does that look like?
Are they all attempting it separately? That means either each gets 2 seconds to try before the next one has a go, which doesn't sound plausible to me, or they're all stepping over each other to simultaneously individually try to stabilise their friend, in which case I'd almost want to impose disadvantage on them for that. However, if we assume that the party are collaborating in their efforts to revive their friend, then I'd say that's what the Help action represents, and thus grants advantage on the one with the highest Medicine skill.