The diag code detects when the sensor is open or shorted and subs the default hot value. Thereafter the EFI code (the routines that determine injection, timing, and idle) use it. The code doesn't "know" whether it's operating off the actual variable or the default, it just goes by what happens to be in that particular register at the time. The default value is simply static.
Remember, most of the reasoning behind EFI is emissions based. A proper emissions strategy would never permit the approach you're suggesting. Nor would it wait until the engine is hot to enter closed loop. The opposite is desired. In fact if you monitor cross counting you'll see it occurs rather soon after engine start, long before coolant reaches anywhere near operating temperature and with or without the sensor connected.
Assuming everything else is working an open or shorted sensor will typically cause opposite of what the OP reported: the engine will falter when cold but run OK when hot. It's because the hot default deprives it of warm up enrichment. It's no different than having a faulty carburetor choke.
Course, it's going to depend on just how cold the coolant is. At 50 degrees it may only run rough for a minute or two but as it gets colder the lack of warm up enrichment will show for a longer period until things warm up. Try it on your car and see...