Besides the US: Turkey, Syria, Russia, Israel all off the top of my head have air force and ground army (along with, at least, a navy) separate under one "Armed Forces" command.

What modern air forces operate as part of the army, rather than separately (as most navies do)?

To properly compare, you need to look at whether the air force has the same level of autonomy from the ground army as the navy does. If it does, it's just like the US. If not, it's not.

The UK has the same high level structure as the US (A “Naval Service” including Navy and Marines, an Army, and an Air Force.)

It's a fairly common structure.

The Chinese PLA actually has more services (Ground Force, Air Force, Rocket Force, Navy, Strategic Support Force.)

There probably are modern militaries where one of the top-level divisions isn't something like an air Force, but they aren't dominant among modern, large-scale forces.