John,
The air gap prevents the waste from being siphoned back into the dishwasher. There needs to be a "space" between the supply and drain. ( Air gap )

The other requirement is that the "gap" must be above the "flood level" of the sink.

Sometimes, the hole in the counter is required. Unless you use one of the three holes in the sink for a single handle faucet without the deck plate,
a spray,
and the Air Gap.

Terry