Not a bad guess if you don't know the flow of things. If outside fan is turning on then the thermostat and wires are good. The same contactor that turns the outside fan on turns the compressor on. The compressor uses a starting cap (fan has one to) to initially start. These go bad all the time. Mine cost $9 on Amazon. https://www.amazon.com/gp/product/B076TKGGGK/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1
On our block we have spares for everything. Was down for no more than 30 minutes. Try to explain to my kids they need to learn this basic stuff. They look at me like I am nuts.
And if the starting cap has gone bad why would it some times or even for a short period go on?
The central A/C goes on only in some parts of the house, and in the other parts of the home it blows very gentle, works as if it's off. Meaning it blows so soft that you wouldn't know the A/C is on, when I go up to the attic I hear the fans and motors working away, and felt the tubing and seems to be air coming into it.
Would think something is blocking but the vents are open and a little air is flowing, thing bad batteries can be the cause for these old thermostats? (the screen in on so wouldn't think the batteries are bad.)