Does a car run cooler without a thermostat?

A vehicle generally runs hot if low on coolant, that is the first thing to check. If coolant levels are ok, make sure your serpentine belt is tight and not squealing. If loose replace the belt tensioner and belt. Third the thermostat may be sticking and needs to be replace, usually located below the top radiator hose where it joins the intake manifold. You give little information on how hot, so it may just be the cooling systems needs flushing and the coolant replaced.

check coolant level in the radiator at the cap
the radiator flow tested for blocked cores
run the fault codes to check for tuning and timing issues
check for extra load on the engine such as low pressure in the tyres, slipping in the auto or travelling at high speeds
check for plastic in front of the radiator
If the thermostat is bad it normally goes to hot ( not a bit hotter than normal) check the cap for correct operatopn

Yes, but it's not recommended. If you have cooling issue, fix the cause, usually a blocked radiator. May be collapsed hose. Colling issues can also come from a blown head gasket.

"rule out" questions - is coolant level OK? Any leaks, especially around the water pump? Has the system ever been flushed. out? If this is the original radiator, it could be clogged. Is the thermostat installed correctly with spring end toward the engine?

Have you verified that system has been "burped" of air? That the fans kick on? Check radiator hoses for kinks. You can try running without the thermostat to see if that helps and can identify if the thermostat could be a problem. You may need to check your coolant mixture too.
Sounds like you have reduced flow or restriction in cooling (heat exchanger) as it runs fine at cooler temp but can't keep up when it gets warm.
New parts don't always equal working fine parts.

check and make sure coolant is at proper level when cold. It sounds like your thermostat may be stuck open. A plugged radiator would make the engine run hotter not cooler. good-day ! with engine running and up to tempurature, touch the heater core hoses under the hood. they should be hot both of them. if not, either your low on coolant. or the thermostat is bad.

CHECK ENGINE COOLANT LEVEL.MAKE SURE COOLANT OVER FLOW JUG HAS COOLANT IN FULL COLD MARK.IF COOLANT OVER FLOW JUG IS EMPTY RADIATOR GOT TO BE LOW ON COOLANT POUR COOLANT IN THE OVER FLOW UNTIL COOLANT LEVEL STAY AT THE COLD FULL MARK.SEE IF RADIATOR DONT HAVE A RADIATOR CAP YOU HAVE TO POUR COOLANT IN THE OVER FLOW JUG COOLANT WILL FLOW DOWN IN RADIATOR UNTIL COOLANT LEVEL STOP AT FULL COLD MARK.IF COOLANT LEVEL OKAY.YOUR THERMOSTAT NEED CHANGING ALSO RADIATOR PRESSURE CAP.HAVE RADIATOR FLUSHED OUT.HAVE ENGINE BLOCK FLUSHED OUT.CHECK ENGINE OIL IF OIL LOOK LIKE MILK SHAKE ON OIL DIP STICK YOU HAVE LEAKING HEAD GASKET THAT WILL CAUSE ENGINE OVER HEAT WHILE CAR IN MOTION.CHECK WATER PUMP MAKE SURE ITS WEEP HOLE NOT LEAKING ANTIFREEZE IF SO WATER PUMP NEED REPLACING.

Hope this helps you. Here are the basics. There are many problems you may run up against (frozen bolts, stubborn or cracked hoses, rusted clamps, etc)that may need to be fixed or replaced before proceeding: SHut off engine, allow to cool completely. Disconnect the cooling fan, remove negative battery terminal connection. Drain radiator by opening draincock and/or removing lower radiator hose, collecting coolant in a drain pan. Do not let it go onto the ground - bad for environment! Open top radiator hose to allow remainder of coolant to flow out. A few quarts of coolant will remain in the engine block. You will want to totally flush and refill AFTER replacing the radiator to get all the crud out of the system. Unbolt radiator and disconnect any other items still attached (transmission cooler, temp sensor wires, etc.). Remove radiator and replace with new, reattaching all items removed from old radiator. Inspect other items and possibly replace suspects. Reattach hoses and make sure the clamps are tightened without pinching hoses excessively. Do a flush job with water/radiator flush, making sure the water runs clear out the draincock. Instructions for flushing the cooling system are usually printed on the radiator flush bottle. Be sure the heater is on and the engine warms up completely to open the thermostat. SHut off engine and drain as much water as will leave. Fill with 50% coolant mix (Dexcool or equivalent). Estimate water remaining in the engine and add 100% coolant to balance. COmplete fill with 50% mix. Remove air by running engine until warmed with the radiator cap off, adding coolant mix along the way.

Half sounds normal however if it starts climbing up to 3 quarters then the thermostat is sticking flush radiator coolant and change thermostat

Mine was doin the same thing when i got it. We fixed the " fan clutch " which apparently sits in front of the radiator and helps cool the AC(according to mechanic). When this got fixed the AC ran cooler and car started running cooler. Like yours, mine was overheating, but running hotter than it should.