crappy coolant "flush" problems

I noticed a little dirt looking sediment in my empty-looking coolant reservoir while I was topping up windshield fluid reservoir. So when I took it to get an oil change I asked if they could do a radiator flush too. I know it must have been low for the overfill reservoir to be empty but I hadn't noticed any overheating of any kind.
When I picked it up, the mechanic said it still needed about a half gallon of coolant since he wasn't able to get to Napa yet (small town, he doesn't keep everything on hand) so I just went ahead to get it myself and top it up. What he didn't tell me was that he was basing that amount off of what he drained out, not the actual capacity (which is 13.2 quarts). Anyway I'm finally able to get the answer that he only put one gallon in after only draining out about 7 quarts and not flushing at all. I've been slowly putting a second gallon while running the engine and heat so that the air bleeds out.
The air is coming out very slowly now, between dribbling barely more in and keeping an eye on the now-normal running temp (after it overheated just bringing it home) was taking a really long time so I just turned it off and put the cap back on.
Is there a gallon more of either coolant, or at this point trapped air in the system? Again, he said he drained out about 7 quarts and put in one gallon. After I picked it up, I've put a second gallon in.


And yes I should have done it all myself- the oil change, the coolant flush, etc but I didn't, and won't make the mistake again, so we can definitely save the time on that part of the discussion!
Oh god is my truck toast. plz help

Welcome to TRS

Systems that use an Overflow Tank are self Purging if any air is left in the system when refilling.

Best way to refill any system is to unhook one heater hose from the firewall or heater valve.
Then start adding coolant to radiator until coolant starts to run out of heater hose and heater core, then reconnect hose, that will get most of the air out.

When you start the engine, and drive it, the coolant circulates and so does any air, the air will get trapped at high points like thermostat, then upper rad hose and upper rad when thermostat opens.
Any air will end up at rad cap opening.
When coolant heats up enough it expands and the pressure in the engine/radiator goes up, when pressure gets to "rad cap rating", say 14psi, then rad cap is pushed open and air is pushed out FIRST then coolant until pressure is under 14psi and cap closes.

The overflow hose comes into the BOTTOM of the overflow tank for a good reason.
And the overflow tank is NOT SEALED for a good reason, it has a vent hole.
Any air pushed out of rad on warm up will bubble up inside overflow tank and it is gone from the system.
When engine cools down the smaller SECOND valve in the rad cap is PULLED open as pressure in the engine/rad drops below -1psi, and coolant from the BOTTOM of the overflow tank is sucked back in.
So air bubbles out, coolant is sucked back in

After an engine cools down the Rad coolant level should be at the very very top, 0 air
If level is low then you have a leak or another problem, like overflow hose leak or bad rad cap seal.

Air is much easier to move than coolant, so as engine cools down air can be sucked back in instead of coolant, from any leak points

Keep overflow tank topped up at its COLD mark and any air will be purged after a few warm up and cool downs
And keep overflow tank clean from debris, if it settles to the bottom it will block returning coolant, so coolant can't be sucked back in, usually you will see collapsing upper rad hose as pressure of -1 to -4psi can't be relieved

Also a general "heads up"
Reverse the heater hoses at the firewall any time you change coolant, or once every few years, this Back Flushes the heater core, pushes out any larger particles the way they came in, it will add a few more years to any heater core's life, and it is FREE

Over time heater core passages can get clogged, and yes that does reduce HEAT available in the cab, but worse is that coolant is no longer flowing in ALL the passages, coolant has anti-corrosion chemicals................with low or no flow those heater core passages will corrode and you get coolant leaking inside the cab, and a HAVE TO REPLACE heater core situation

Great write up RonD ..Thank you

Thanks so much for the detailed response, I was panicking a little wondering what happened.

...and yet, the shysters get away with their scams! How do people live with themselves?