What year and engine?
You can pass water through a rad while it's in a vehicle, but that won't really "flush" it, the passages are too small, you really need to pull it out and run water through it backwards, back flush it, pushing out any debris the way it came in, but that is also just a stop gap if tubes are starting to plug up, you can lie it flat and fill it with a warm CLR mixture, then back flush it, repeat a few times.
With rad out put it on it's side, so tubes are running up and down.
Put hand over lower hose outlet and fill rad using upper hose outlet
Put on rad cap if needed.
With rad full stop filling and remove hand from lower outlet
Water should drain out instantly, listen for any gurgling sounds from upper hose outlet, air being sucked thru partially blocked tubes.
now the test...
All water should out of the rad now, flip rad upside down, if ANY water comes out it could ONLY come from BLOCKED TUBES, a little water or a lot, doesn't matter, any blocked tubes means rad needs to be replaced.
Other rad tests.
With engine at operating temp, t-stat open and rad cap on, feel upper and lower rad hoses, lower should only be a little cooler, not a lot cooler.
Squeeze the upper hose and lower hose, lower should only have slightly less pressure, if it is much easier to squeeze lower hose then rad has blocked tubes.
Coolant is sucked from lower hose by the water pump and pushed to upper hose, so if rad starts to get clogged up the coolant in the lower rad can't be replaced as fast and lower pressure is the result, so easy to squeeze lower hose.
Shut engine off and move fan out of the way so you can run your hand across the rad surface, feel for cool spots, those are where there are blocks in the coolant flow.
A good Rad should have nice even heat across all surfaces.
On some cooling systems the heater core is used as the bypass for the water pump, this means if the heater core starts to clog up it will cause the engine to overheat, it will cycle up to hot then back to normal, almost randomly, not usually related to engine load.
You can by-pass heater core to test this.
Turn on heater while driving and see if heater temp matches gauge, as engine temp goes up does heater temp go up as well?
If not the water pump may not be working as well as it should, worn or slipping impeller blade.
If heater gets cooler then air is in the system, could be from a leak in the system or the beginning of a head gasket issue.
A t-stat controls the lower operating temp of an engine, and unless it is stuck closed it would not be the cause of overheating.
I would drain the new coolant for reuse, and run water until problem is solved, you can just remove the t-stat at that time if in doubt, engine will take longer to get to normal operating temp, but that's all, that is what the t-stat is for, getting engine warmed up faster.
Running water for a few days or even weeks(in the summer) won't hurt anything, and no it doesn't have to be distilled water, if you can drink the water and it doesn't taste "funny" then it will be fine to use in any engine.
When you replace a t-stat you should usually replace the rad cap, that's just a general "rule of thumb", both are cheap
These are just general cooling system checks.
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