That tree isn't a terribly technical take down. It's going to be a high impact job but they're ripping out the yard anyway. There aren't any electrical lines so he's OK there. No obvious signs of decay/compromise. The fence needs to be considered, but if the homeowner was considering $1400 before (and I assume Anthony's under that) he should still be making enough to cover whatever damage he does to the fence. And I'll bet mediocore money he's going to damage the fence.
He mentions he owns a climbing saw so I assume he's been climbing and removing limbs for a little while. I'm a bit concerned that he didn't know how to tie into a spar, but he's never done a takedown before. He didn't mention, nor specifically respond to queries about a groundman, but I'm assuming he'll use one and will be roping down a few pieces. Sure, he's gonna cut leaders he'll wish he had later for lowering limbs, but that goes with inexperience and it's not the end of the world.
If it were my job would I turn him loose on that tree unsupervised, Hell no. If it were my tree and I knew Anthony had never done a take down before would I let him do it to save a few bucks, Hell no. But it's not about me, it's about a fella whose nickname and signature suggest he fancies himself some sort of tree bull. I think he'd make a great climber candidate if he could get with someone who would tutor him a bit. But my impression is Anthony has decided to do this tree. And if he's been sawing in trees before, he's rigged stuff down, and he takes his time and uses his head more than his saw, he'll be fine. This tree will be a challenge for him. And if he gets it down without incident, he'll gain a lot of confidence and he'll be one step closer to being NY's Saw Boss.
Good Luck,
BAB