AC not turning on until well over 4 degrees of set temp

If a repair fixes a problem, then in general one does not look for more things to do. Wait before you do anything more.

In my experience the 240 V power line goes directly to the condensing unit outside. In the US there is usually no 240 V line to the air handler inside. In the US most residential air handlers use 120 V on an entirely different circuit from the 240 V line to the condensing unit outside.

There is a "relay" (called the contactor unit) inside the condensing unit which switches off and on the 240 V high current power to the compressor and fan. Failure of the contactor unit is common, but at present you have not given any reason to suspect yours should be replaced.

EDIT

But in Europe there is only one voltage 230 V to 240 V so the air handler would be at 240 V. I assume that the air handler and the condensing unit would still be on different circuits, but I really don't know.

Yes, there are energy efficiency code standards for residential buildings. The Code is divided into 1) one and two family structures, and 2) three or more residential structures. (There are also energy standards for manufactured housing, log homes, commercial, institutional, etc. too, but that does not apply here.)

I will reference one and two family structures only.

Each building department “adopts” their own “code”, but this is recognized throughout the U.S. This is based on the International Code Council (ICC), 2017 edition, which is the current edition.

First, you should know that the Code provides a “proscriptive path” for all building elements. If you use them, then you are approved for construction.

Second, the Code allows substitutions and “alternate means of construction “, but the calculations required to show you meet the Energy Code is difficult (and would probably require a mechanical engineer to do the trade-off calculations.) That is to say, you don’t have to use R-21 insulation in the walls, but if you don’t, you’ll need a more efficient heating/cooling system...etc.

So, here is a list (you can verify this list by calling your local jurisdiction...Building Department) and it doesn’t matter if you’re over a garage or not:

1. Walls: R-21
2. Flat ceilings: R-49
3. Vaulted ceilings: R-30
4. Under floors: R-30
5. Windows: U-0.30
6. Window area limitation: n/a
7. Skylights: U-0.50
8. Exterior doors: U-0.20
9. Forced air duct insulation: R-8

Note: R-values are for insulation ONLY. It does not include values for siding, sheathing, etc.

10. Gas fired furnace or boiler: 94% eff.
11. Air source heat pump: HSPF 9.5/15.0 SEER cooling
12. Ground source heat pump: COP 3.5 or Energy Star
13. Ductless heat pump: HSPF 10.0 in primary space.

There are many other Energy Code requirements like : window air leakage, slab on grade performance, etc., but too numerous to list here.

SUMMARY: If your contractor met the above requirements, I’d expect better performance in your living space above the garage.

If the space is not too cold (in heating season) or not too hot (in cooling season), then the heating system is of adequate size.

I’d then check thermostat. Some have a “dead zone” setting...usually about 3 degrees. That is to say, if you set it at 70, then it must drop (or rise) 3 degrees before it turns the heating/cooling system on. If the thermostat is located in a drafty area (say by an exterior door) it could trigger the thermostat.

I would then check insulation thicknesses. One easy check is to see if the walls are made out of 2x4’s. The required insulation cannot be installed in 2x4 walls...a 2x6 wall is minimum. (You can easily check thickness of walls by looking at the walls by the windows.)

I’d check the ceiling insulation, because heat rises and you’ll quickly loose heat through the ceiling/roof quicker than walls. (Likewise, If cooling is an issue, then check the floor insulation. )

If the windows are not double pane or the exterior door does not have weatherstripping, then they do not meet minimum requirements.

Remember, your large temperature change is caused by a lack of retaining your heating/cooling AND your heating/cooling system not responding quick enough.