Lighting – Recessed light holes

Unless there are other restrictions such as clearance to framing members or insulation or open space above them you can use them. "New construction" is when the framing is open and not covered by drywall^*; the fixture is typically attached to a framing member. As opposed to "old work" which is where it's closed on both sides and you're going through a small hole in the drywall; the fixture is often attached by clamps to the drywall itself.

^* With some "new construction" fixtures you need to have the wall open on the side you're working; with others you can get away with having just the other side open because the fixture will give you enough room to work in.

Wow, I'm surprised to see that an electrician did this to run wires! That number of holes certainly looks excessive.

While the method you suggested will work just fine, it seems like it would be quite repetitive. It might just be easier to cut out a rectangular strip and replace it with a new piece of drywall - then tape, mud, sand (and repeat) and paint. You might have to paint your entire ceiling in order for it to blend well.

If I were doing this (and I installed 30 of them in my house without cutting any extra holes in the ceiling), I would have cut the holes for the can locations and then fished the wire using a 6' flexible drill bit and an extension bit if necessary. An inspection camera comes in handy here too to help you see what's going on. If I was really stuck, I'd start by cutting a small hole only for the inspection camera (maybe 1" max - easy to patch), and if it really required a large hole, only then would cut a large hole so I could determine where the problem is.