How do I stop the coolant overheat message

I have an issue, when I stuck in traffic my temperature gauge jumps from halfway to max with the stop light flashing and beeping at me and message comes up coolant temp too high.

After replacing the temperature sensor I realized this is not the cause.

However, after flushing the radiator I realized that the radiator fan is not kicking in until it is already saying stop too high,

What is causing this delay the radiator fan should kick on way before this ⚠️ warning?

I feel it is the thermostat, but as a new driver and know nothing about cars I would really like some advice.

Thanks in advance