"I refuse to have a debit card because, unlike credit cards, you have NO PROTECTION."
WRONG.
Absolutely 100% WRONG.
The protections you have are outlined under the Electronic Funds Transfer Act.
The levels of protection are different between the two types of cards and are outlined in this article:
https://www.identityiq.com/scams-and-fraud/credit-card-fraud-v-debit-card-fraud-are-you-protected/Credit card transactions and protections are governed by the Fair Credit Billing Act, which is also outlined in the article above.
As for why the "teller" at your bank won't use the bank's bill payment service? The only reason I can think of is that she's a stupid panic monkey who's afraid of technology and who thinks that paper money is actually a sign that Satan walks the earth and possesses us all.
The process for you paying a bill by writing out a check and sending it in is, in reality, NO different than you clicking a few buttons on your computer. Often the bank's bill payment service literally writes a check and sends it to your biller (for smaller companies that don't do bundled electronic transactions).
Online bill payment is generally safer and more secure than writing a physical check and mailing it out because of the multiple layers of encryption and protection -- your computer (if you're keeping your software up to date) your bank's, and the processor. And, in the case of full electronic bill payments via bundled transactions (larger companies, utilities, etc.), THEIR encryption and protection.