The majority of dealers I have worked for sent a crew to help deliver, and left a tech (me) to fill, connect wiring, install filters, cover lift, cover locks, set up steps, install handrails (if you got them), install optional ozone (if you got one), and give you "spa school" after it filled. Put the hose in first and it's usually full by the time you're done cleaning up.
Others who would not connect electrical or used a spa mover for deliveries would send a guy (me), after you got it hooked up, for spa school. But you still have to install the extras either way, which takes a couple hours for one guy whenever you do it. It makes little difference if it's full or empty when I start, assuming you have decent water pressure. But you are correct that a 3-man crew will generally not wait for it to fill. 3 experienced guys can knock out the installs in a half-hour or so and, in that case, would be watching the tub fill, on the clock, for quite a while.
I'm not saying there aren't exceptions, just that it's not the norm, in my experience, for reputable local dealers to drop and run. We joke about being married to them, not forget them as soon as they hit a pad.
Now anything drop-shipped, like homeshow, parking lot or tent sales, and online orders, are an entirely different story. These are often delivered by a hot-shot (contract mover) who hauls whatever for a living and doesn't know the first thing about a hot tub and may not even be aware that's what they put on his trailer. He also may not be able to remove it from his trailer without you hiring a local spa delivery crew to come take it off for you. I have heard of them being pushed off in front of the driveway in those circumstances, and have been the guy called to remove them more than a few times. But that's still a very small fraction if the tubs out there. Though it is one of the reasons I recommend avoiding such sales.