If sounds like you are asking one question but looking for an answer to another.
You question of what a thermostat does is: Controls the temperature point which the coolant is allowed to return to the radiator. I know this is an over-simplification but serves the purpose of keeping the focus on looking for the real problem. Which is why your cooling system is not controlling your engine temperature.
We need to define the parts associated with the engine cooling system and comment on their contribution to this process.
1. Radiator (better defined as a water to air heat exchanger)
2. Water pump. used to move coolant through the system.
3. Thermostat. Already defined.
4. Water pump drive (pulley's and ratio's)
5. Coolant (water and additives such as
antifreeze)
6. Pressure cap.
7. Hoses
8. Waterpump to engine bypass. (The small hose usually running between the intake manifold and the water pump.)
9. Fan: move air through the radiator.
Possible reasons for over heating:
1. Radiator's core cross sectional size: Basic rule of thumb is 1 square inch of surface area to 1 cubic inch of displacement. Or as related to HP: 1 Hp to 1 square inch of surface area. If you have less than this area you may not have enough cooling area.
2. Radiator thickness: This has more to do with capacity than cooling ability. Too thick a radiator can be as much of a problem as too thin. REMEMBER a radiator is a water to air heat exchanger so if you cannot get air through the thick core you cannot cool the water. Good rule of thumb is no more than a two inch thick core for aluminum or 3 core for brass.
3. Water pump. Most stock pumps are more than up to the job. Especially Ford units. I would not think your problem is here. Be aware that some after market units perform well at upper RPM figures but actually flow poorly at normal street RPM. Therefore they can contribute to overheating at idle and low RPM running.
4. Thermostat: Not usually the cause of overheating. Unless stuck and this rarely ever happens and is quick to diagnose.
5. Running without a thermostat: Bad idea. Does not allow you to stabilize your running temperature. The engine does not get a consistant coolant temp. Can lead to overheating but rare.
Oil sluging and excess ring wear. Besides, how do you control HP without temp control?
6. Water pump drive: Often ignored issue. You got to move water to cool the engine. Basic rule for performance engines is a 1:1 ratio crank pulley to water pump pulley.
7. Coolant volume: Air in the system. excess air (a common problem with low mounted radiators like Cobras) create hot spots. Fill your system to the very top then run engine and let the excess blow off through the 13lb. cap. The balance should be fine.
8. Fan air volume: VERY IMPORTANT. And I believe is your problem. Remember the radiator is a water to air heat exchanger. And without adequate air movement you cannot cool the coolant. I am going to assume you are using some kind of electric fan as most Cobra users do. Almost all electric fans offered that will fit a Cobra will not move enough air to cool a high performance engine at low speeds and idling. THE RULE OF THUMB IS 10 CFM PER CUBIC INCH OF DISPLACEMENT. In other words if you have 350 CID you need 3500 CFM. THe highest common fan rating is 2900 CFM.
9. There are other potential issues that can cause overheating but they do not usually contribute to a problem that allows the temperature to rise as fast as your question suggests. So don't concern yourself with these now.
Back to testing the fan issue.
Get a infrared temperature gun. (Usually available at most part stores and around $100.00) Start your engine and allow it to warm to thermostat opening temp. Use the temp gun and point it at the top of the radiator where the hose enters the tank. Take a reading against the tank as close to the hose as possible but not on the hose. Record the reading. ( Should be around 10 degrees cooler than thermostat opening point) Now let the motor to start to heat above the thermostat temp rating. Again measure at the top THEN take another reading where the lower hose meets the tank. ( Top should be very close to engine temp reading and lower hose shoud be aound the same) Record the readings. Turn on your fan and let the fan run for 2-3 minutes while allowing the motor to continue to heat up. Of course don't allow your motor to exceed 230 degrees. Measure the temp at the top and bottom hose points. IF THE TEMPERATURE DOES NOT DROP DOWN AT THE LOWER HOSE BY 30 DEGREES YOU ARE NOT PASSING ENOUGH AIR THROUGH THE RADIATOR. If you find that this is your problem there are ways to help solve this and you are welcome to contact me via e-mail or phone @ 541-447-1398.
Good luck.