Thermostat location and replacement?

Only matters if the tstat mounts on the side of the block. The valve should face up to allow air to bleed out of the system. If your cooling system and heater work OK, dont bother with it. Older tstats never even had em.

It is possible to get a new one that does not work or that is the wrong temperature.

How hot are you letting it get? That is a 192 degree unit, If you test it in hot coolant it should start to open at 192 and fully open by 212 degrees.

It is also possible that the water flush has frozen in a small spot and the pump cannot circulate the coolant past it. Not really likely unless you took a couple hours from flushing to filling.

What was the original reason for the flush? No heat or overheating?
If you fill the system and just start the engine, does the coolant start bubbling or running over? If it does and the issue was overheating you may have a bad head gasket. Especially if it goes from cold to hot in a very short time.

Install a brand new thermostat.

Since there is no bleed valve, jack up the front end on the side where the radiator cap is until the radiator cap is higher than the rest of the cooling system which includes the heater core, then remove the cap, top off the coolant, start the engine and allow it to warm up for five minutes or so, Shut off the engine and then grab the top radiator hose with both hands push it down as far as you can then squeeze real hard with both hands and force any air out the filler neck. Top off the coolant, leave the cap off for now, start the engine and let it idle and make sure the thermostat opens and the system does not overheat before putting the cap back on and lowering down the vehicle.
You could also just pull that one front side up onto a curb also.
If this does not work, then install a new thermostat cause I learned long ago from an engine re-builder who told me once an engine overheats it ruins the thermostat. I suppose though to what degree it overheats just how much it ruins it though but it can be true.

Temperature gauge moving half way will not normally be overheating. That would be about a normal area for the thermostat to start opening. What you need to do would be to actually measure the temperature to see if it is okay or actually running hot.