About 10 business days after I initially submitted the payment and paperwork for a 30-day tourist visa extension, I returned to the DAS office for a new stamp in my passport.
Aside from a 20 minute wait, there were no hiccups in the process. The drag, of course, is having to repeat the visits every few weeks.
At least in the future, I will only have to submit a payment and bring my passport.
The question I still have is whether that payment always needs to be deposited into a bank account, or whether I can pay at the office.
The latter would make life easier, so I am guessing I will have to do the former!