I don't think it's the security of receiving payments from the bank as much as it's the financial return of it. If you're going to pay over time, they want as much of the money as possible with minimal processing losses.
ACH, what they very likely use for bank withdrawals, has processing fees in the ballpark of 1%.
Gift cards are sold by merchants to stores at a discount, typically around $85-90 for a $100 card. The store then sells it for close to $100 (or, sometimes, with discounts as many here target).
But this means Disney only ends up getting 85-90 cents on the dollar (or however much they sell their cards to retail stores) for this money. If you're turning around and spending it on dues, that's essentially a processing fee of 10-15% for Disney to absorb into maintenance fees somehow.
For reference, credit card transaction fees are closer to 3%.
In light of the processing/transaction losses of gift cards, I'm surprised they let dues be paid by gift cards at all.