I think this is in interesting discussion. It seems like I may be a middle-of-the-road guy.
I set up the direct withdrawal for the current year owed. I consider that as part of the e-file process as much as setting up Direct Deposit.
But I don't set up electronic Estimated Tax payments. There seem to be too many things that can happen between filing and January 15th (clients forgetting about the withdrawal, changing bank accounts, income/circumstances change, etc.) and I don't want to deal with those problems. So I give the clients the vouchers, point out IRS Direct Pay and EFTPS, and let them deal with it (and let them know they need to tell me the amounts and dates of payments for next year).