A/C blows hot air after engine coolant service

Hello,

Can the engine coolant system/fluid directly or indirectly affect A/C performance? Or can an engine coolant system service affect (good or bad) A/C performance?

Backstory:

Three weeks ago, did some 100,000-mile maintenance on my FJ: Engine Coolant system service/fluid flush, Transmission Fluid Flush and replaced front/back shocks. After I got it back, my A/C only blows cold air at night (60 degrees outside). And even at night after it’s been running a bit or on the freeway, it’s nowhere near as cold as it was before the service. During the day it’s worse (70 degrees outside). I can just barely feel cold air if I put my hand right next to vent and put the vents setting to blow only up top. Before the services I mentioned above, A/C was blowing cold no matter how hot it was outside. It would even blow cold when idling but blows very warm now when idling.

I didn’t notice this problem till about two weeks later. After I got it back, had a cold spell outside and wasn’t using the A/C. But had a couple warm days in a row this week, used the A/C, and then realized the issue. However, it was very warm in the days leading up to the service, so I had used the AC right before I took it in. (Live in So Cal by beach and in winter it can be 80 degrees one day and 65 the next).

When realized issue, called repair shop right away who did service. Was told the engine coolant service/fluid flush-replace has nothing directly or indirectly to do with the A/C, that it was just a funny coincidence. So, they want to charge me again ($152.00) to check AC/recharge it and add “Freon” if it’s low. And more money if there’s a bigger issue (leak, etc.).

My Question(s):

Is the repair shop being truthful that the A/C problem is “purely coincidental” and has nothing to do with the engine coolant system and service they did?

He also told me the “Freon” for the A/C is a separate fluid than the coolant for the engine. Is that correct?

During the engine coolant service, could/should it have been discovered that the A/C system had leaks?

If A/C “Freon” is leaking, would there be a puddle under the FJ after it sat for a while?

If I don't turn on A/C, how long can I wait to get it fixed? I don’t have the money for repairs, so might be about a month before I can.

Things I’ve tried:
Some experiments on the A/C as described above (when it’s blowing all hot air, temperature of day, changing vent settings, running the car on fwy, etc.) Nothing I try causes AC to blow at the cold temp like it did before service.

Last thing I’m going to try is changing the cabin air filter... It’s due. I haven’t tried that yet before posting because don’t have the funds to buy one right now (was also going to try to change it myself). But if it was the cabin filter, I logically would have had this problem before the service(s).

Below is the engine coolant service portion of the repair ticket at service was completed:

Coolant System Flush with Extended Life Coolant.
-Flush entire coolant system.
-Check for any visual coolant leak.
-Check drive belts and upper and lower hoses.
-Test drive for proper coolant system temperature.

Thank you all in advance!

FYI: I have no automotive repair experience - so am seeking advice on whether repair shop was truthful and then I can decide whether to take it back to them. I have only taken it in to them once before for spark plug replacement and had no problems.

Christine