Bleeding the cooling system?

Easiest way is to put the front of the car off of the ground so the radiator fill cap is higher then the heater core, start the vehicle and run it at about 2000 rpm's and keep villing the coolant untill no more bubbles appear.

The idea is to get any trapped bubble out of the system which can be done by lifting the front or the rear of the car while filling, it can help. What problem are you having?

You don't have to run it until it's hot, you just start it and let the pump circulate the coolant at 1,500-2,000 RPMs. Then shut it off and top it up. Then put the cap on and run it until the engine is warm and the thermostat has opened. Now let it cool off and check it, if it's low top it up. As for boiling over, it could without the cap, but most vehicles won't do that at an idle because the engine isn't generating that much heat.

Okay, the cooling system starts off full, radiator full and bottle at full cold. You start the engine, and it heats up. As it does the coolant will expand as it heats up. (Has to do with how the water absorbs the heat and makes the molecules speed up and expand to store heat, then they hit the radiator and dissipate the heat out into the air) Now you have HOT expanding liquid. Liquids don't compress, so unless you want that full radiator to split open you need a place for it to go. On modern cars it pushes the spring on the radiator cap open and releases some pressurized hot coolant into the overflow reservoir or expands in the pressure bottle. So now the engine is fully hot, and the coolant is doing its job. You get home and shut the engine off. For a bit the engine will actually cause the coolant temp to rise even more because there is no coolant flowing. That ends and the engine starts to cool down. The coolant starts to condense as it releases the heat. As it does it starts to leave a void inside the radiator and engine. That void is a vacuum point. In a vehicle with a reservoir this is the point that the second valve in the cap opens under that vacuum, and it pulls coolant back into the radiator. This does two things, one it keeps air out of the cooling system. Rust and aluminum oxide inside the cooling system will destroy an engine very rapidly. So, to prevent that air entry into the system to help prevent that corrosion. It at the same time removes the air to prevent any from being drawn into the engine and causing a hot spot in the engine due to the lack of water flow through the air pocket.
Now in a vehicle with no leaks and a properly operating cooling system any coolant lost would be a miniscule amount through the secondary overflow tube. So maybe over the course of 2 years it might evaporate a teaspoon. Any coolant loss over that means you have a leak somewhere.
If you are saying that you top it up cold. Then drive it around and have to add coolant again within a couple of days, you have a leak somewhere. Either external, like a pinhole in the radiator that leaks out hot coolant that evaporates in the air, or internal, IE a head gasket that is starting to fail.

Once the engine has run through a couple of hot/cold cycles the coolant should stay at a constant level, unless you have a problem. Take a look at where the overflow connects and make sure it's sealed. If you still need to add coolant after a few heat cycles, it's going somewhere. A compression test may not show it either. Want to see an interesting but technical explanation on that. Visit the YouTube channel Schrodingers Box, look for a 3-video series on a Jeep, the first one starts "did liars and thieves misdiagnose this car" Isn't the first time I've seen similar things.

It's a good thing if the thermostat is open but the system is designed to be "burped" with it cold. I don't bother these days as I use a vacu-fill on most vehicles. So much simpler and you can use it to test the system as well. For head gaskets I've been known to pull out a few tests before condemning the engine. I'll run a leak down test then a cooling system pressure test, a liquid gas test and even stick a 5 gas detector in the coolant overflow. This shows a few of them. https://www.2carpros.com/articles/head-gasket-blown-test
It's not like the days when you could pull the heads off a small block and swap the gaskets out in a lunch break so I want to be sure beforehand.

Hmm, that last image with the piston down in the bore isn't promising, the yellow on the cylinder walls and the clean piston surface looks like coolant has been cleaning it. The coolant will get into the cylinder and the ignition cycle will blast it into steam. That steam will break up and remove any carbon or deposits from the combustion chamber. So, if you look in and see 7 cylinders that have some carbon and ash on the piston tops and valves and then find one that looks clean enough to eat off them, you have a problem.

So, when you installed the new radiator did you remove the hose connection covers before installation? They should be red or blue.

Likely those are for secondary overflow or bleed lines or if it's a true race radiator they could be for a push feed coolant fill system that allows for the race cars cooling system to be topped up even with a hot engine.