There is no reason that I can think of why running without a thermostat should cause overheating problems. It is simply a mechanical device that throttles the water flow out of the engine. If you take it out the water will circulate at max rate and cool the engine excessively. So, without the thermostat you would expect the engine to never warm up fully - if everything else in your cooling system is working correctly - and run cold / cool.

If it is overheating without a thermostat, it sounds like you have some other issues. Can you confrm that the water is circulating through the radiator when the engine runs? The hoses to and from the radiator should warm up quickly when you start the engine from cold as the thermostat will not be stopping the flow. One possibility is that one of your radiator hoses has collapsed internally, resulting in a blockage.

Seems a coincidence that the problems started with the car overheating initially. Maybe there is a pre-existing problem that has not been cured by the repairs. I cannot believe there is anything other than a lack of coolant flow or quantity that would result in overheating - presuming that the radiator and cooling fans are working normally.

Wolfgang
Wolfgang