Draining and refilling a cooling system is not hard to DIY, but you must be responsible with the coolant - it’s poisonous to anything that drinks any.
I would drain the radiator (and block, if it has its own drain with the heater set to Hot. It doesn’t have to be running, but the Hot setting is needed on many vehicles to open the coolant circuit to and from the heater core. Then refill with plain water (maybe with a cooling system cleaner, too) and run it for some minutes or hours. Repeat the drain and refill a time or two, using fresh water especially if you’ve used a chemical. When it’s clean water coming out, then put in the correct amount of antifreeze.
This is all described in many DIY books, but the above will get you started.
Of course, after a while you’ll be checking to see if that foamy stuff is back.
Some antifreezes react like that to air. Maybe your coolant recovery tank’s hose to the radiator filler neck, or the rad cap, is letting air in.
Yes, of course the cap is letting air out and air in to make sure the cooling system holds it designed pressure. In order to do that, it has to let air out to compensate for the expansion of the coolant, but not exceed recommended pressure. When the engine cools down, air will have to be let in as if not, the hoses would be sucked completely flat by vacuum.
If the recovery tank has enough fluid, and its pipe and hose connected to the radiator are intact, there will be no air expelled from or drawn into the cooling system. The only air contact is inside the tank, where the rising and falling surface of the pool of coolant is open to the atmosphere.
I think that we can all agree, that the recovery tank is a reservoir and to keep the main cooling system void of air. and I agree that the coolant in there will be the first to be exposed to air, but that also implies that the brake fluid in the brake master cylinder and further down the system will not be be affected by it’s hygroscopic ability as what is on top of the master cylinder is also a reservoir. The diff. from new to worn out brake pads and back to new should be negliable.
Nevertheless, this has nothing to do with OP’s breached headgasket.
Btw. I own a car with 2 hoses and cirkulation through the recovery tank, the small one is there to expell air after a refill of the cooling system.