In the case of the hose leak, the gauge may have not read hot because there
was then insufficient coolant in contact with the sensor. Unfortunately,
running hot can damage the sensor. I haven't seen this much on Hyundais, b
ut it does happen. And worse, considering your description, the engine may
be damaged from overheating.
You can check the sensor itself by comparing its resistance to the specifie
d resistance at the temperature of the coolant. You can also attach a scan
tool and read the coolant temperature from the engine control module (whic
h uses a separate sensor element) and compare it to the gauge reading. You
'll probably need to take the car to a shop to accomplish this method.
Presuming they are both functioning, the easiest way to check the water pum
p and thermostat is to check for coolant flow. Start the engine cold with
the heater and a/c off. The radiator hoses will be cold. As the engine wa
rms up, the upper radiator hose should get hot first. As it reaches proper
operating temperature, the thermostat will open, allowing coolant flow. A
t that time the lower radiator hose should be as hot as the upper radiator
hose. The fans should start running to manage the coolant temperature. As
long as the fans continue to cycle on and off (slow speed, not high) while
the vehicle is idling, it's a reasonable assumption that the basics of the
cooling system (thermostat, water pump, fans) are working properly. The f
ans coming on indicate the ECM is attempting to keep the coolant temperatur
e low enough, and the fans turning off indicates that the coolant did indee
d cool down a bit.
There is no thermostat in the radiator cap. It releases coolant via a pres
sure valve. Most are around 1 bar. It allows coolant back in from the res
ervoir via a valve that opens when a vacuum is created in the radiator.
The torn gasket on the cap can cause a leak. Even if it's not leaking now,
I'd replace the cap. It's too cheap to take the chance.
Nonetheless, I stress that things under the hood being hot is not an abnorm
al condition and does not necessarily indicate an overheating condition.