Our quote says in plain words whether it covers the sawing only or hauling the pieces away too. Sawing and removal are separate jobs, and some cutters make the outline cuts and leave the cut slab sitting right where it was. If you're comparing quotes, check that line on each one first, because two prices that look far apart can turn out to cover different work.
We scan before we cut to find rebar, tensioned steel cables and live wiring buried in the concrete, because hitting one turns a simple cut into a repair. We cut wet, since water on the blade cuts the dust down a lot, and that matters most inside a home. Before we start, we agree with you on where the slurry, the gray mud the saw makes, gets collected, so it's contained instead of washed away.
A lot of cutting comes right before a pour. When part of a drive has sunk or a patio has cracked through, we saw out the bad piece and build the new section the same way we'd handle a new concrete driveway or concrete patio construction. That means most of the time goes into the ground under it, not the surface on top.
The base is the part nobody sees on day one, and it's the first thing people want to trim from the budget. I'd keep it. Most cracked and sunken slabs fail because of what's underneath, not the concrete itself, so I'd rather dig out and pack the ground properly than put a nicer finish on soft dirt.