Large shoe is actually more advantageous to card counting, and mechanical shuffling doesn't make a difference (but the frequency of shuffling does).
It's perceived the smaller shoes are better since they are easier to county, but with a smaller shoes the count can't swing as far one way or another, so with a large shoes there is a high probability of getting better odds.
The main thing they do is not play through the whole shoe, i.e. they shuffle before they get all the way through. This prevents the count from being allowed to swing as far as possible.
It's not just that, though. They will also shuffle if you substantially increase your bet, or if you're a new player coming in with larger bets. So even if you get a good count you won't be able to exploit it.
It's perceived the smaller shoes are better since they are easier to county, but with a smaller shoes the count can't swing as far one way or another, so with a large shoes there is a high probability of getting better odds.
The main thing they do is not play through the whole shoe, i.e. they shuffle before they get all the way through. This prevents the count from being allowed to swing as far as possible.