To answer your question, the caps being loose would not cause the substantial loss of refrigerant you describe. Their main purpose is to protect the valve, keeping dirt and other contaminants out.

The technician may be "guessing" there is "probably" a leak and he should look for it. In fact, a leak that substantial, he is obligated to look for it.

I'm not a fan of adding any monkey snot leak sealer into a unit. It causes more problems than you want and doesn't really seal anything. I personally don't introduce anything into the refrigerant circuit except refrigerant. I agree with you, the technician should take a leak detector and look for the leak, it is a relatively easy process and a leak that size should only take a few minutes.

However, I'm assuming he just made a guess in response to your call and will check it out before introducing anything into the system. If you think you can convince him to use a leak detector instead of red die, try to do so. If it is a large leak in the evaporator coil, that stuff is going to blow all around inside the plenum where the coil sits.

The size leak, if indeed it is a leak that is your second problem, would leave traces of oil. If you didn't see any evidence on any of the lines or where they connect, then it could be in the evaporator coil or, somewhere in the condensing unit. It is a very fine oil so look closely.

Or the tech could do an isolation test Sealing each part inside from the outside pressurizing and waaiting to see which one leaks time consuming and cost money then after locating which component is leaking fix or replace as needed. As far as putting sealant in the system and dye he's just trying to ge0t another 3 or 4 hundred out of you cause that won't work.Call another tech.

We stopped adding refrigerant years ago. We can remove whatever had been in the system. We start with fresh refrigerant. We dont charge our customers 15, 20, 30 or 40 dollar a pound for refrigerant. We double our cost. period.
That a 200% mark up. We pressurize our systems with nitrogen,
and have a flat fee based on percentage of pressure used. We bring our systems to 500 microns..with a 3 minute creep upward to 600. We weigh in when we can the refrigerant. We have a labor rate of 125.00 per hour. If you come to a system with a leak. Fix it or get out of the business. We use bubbles, electronic leak detectors, glow in the dark :), human eyes, we even pressurize coils and put them in a tank of water...We STOP Leaks and fix the system. If you want to use a sealant, well try it. Am sure 10s of thousand have it in there systems today and are comfortable.