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.

Although contests are RAW, how to run a contests is explicitly made the DM's responsibility via their own good judgment of the specific situation at hand. That means that, RAW, as soon as a contest is on the table (so to speak), it's in the DM's hands to make any and all adjustments necessary for it to make sense.

In this way, the rules are — again, by RAW — not telling the DM how to do it, only giving them a basic tool to riff off as necessary. Some situations will call for using that judgement to give a stunned character the chance to oppose a grapple, while most situations will probably not.

It might have been nice to have corner cases like this nailed down instead of punted to the DM, but relying on DM judgement is intended to be a feature of 5e, so this is quite intentional. DMs are smarter about the situation in front of them than a rulebook could ever be, and 5e's design intent is to harness that constructively.