How to get the air out of the cooling system

You aren't opening the system where the air is trapped. That is under the thermostat. Even though the engine has warmed up, for a thermostat to open, it has to be hit with hot liquid. Hot air won't do it.

Most import engines have thermostats with tiny bleed holes in them to allow air to sneak through and go to the radiator. Many replacement thermostats don't have that bleed hole, so you have to manually bleed them. Look for a bleeder screw on the thermostat housing. If you don't find one, look for a sensor or a threaded plug on or near that housing that you can remove. If you have to remove a sensor, you'll need to unplug it. Do that with the ignition switch off to avoid setting a diagnostic fault code that could confuse someone in the near future. Fill the system until coolant runs out, then reinstall the plug or sensor.

I'm not familiar with your specific engine but thermostats are rarely next to the water pump. Follow the upper radiator hose to the housing it attaches to and look there.

If you're seeing air bubbles coming out of the radiator, you have one of two things, Either air is still self-bleeding and will continue to do that for a while, or the cylinder head gasket is leaking. Does the engine overheat? Did you replace the water pump because it was leaking, or to solve some other problem?

If the engine doesn't overheat, drive the car at highway speed. The higher water pump velocity will help push air out of the heater hoses.

Your mechanic can perform a chemical test at the radiator to check for a leaking cylinder head gasket. It only takes a few minutes. The test involves drawing air from the radiator, while the engine is running, through a glass cylinder with two chambers partially-filled with a special dark blue liquid. If combustion gases are present, the liquid will turn bright yellow.

Here is how to do the test becasue it sounds like a blown head gasket.

https://www.2carpros.com/articles/head-gasket-blown-test

Please let us know what you find. We are interested to see what it is.

Cheers, Ken

The head gasket test is only about 80% accurate so you could still have a blown head gasket. If bubbles are constantly being generated then you have a blown head gasket.