I'm not exactly sure what you mean when you say "Stores don't have to deal with the volatility since it's the 'problem' of the payment processor."
That doesn't really make sense to me. Are you saying that if Amazon is selling some 100$ item and decide to take bitcoin for it, and the next day that dollar amount is worth 20$, that it's the payment processor's issue? I don't see how that really relates. It would seem like Amazon is the one taking the 80$ hit in that situation because they used fiat to purchase/obtain the item, and then are selling it to someone for bitcoin.
You don't get it. If amazon sells something worth $100 in Bitcoin through a payment processor, Amazon receives the $100 in fiat (minus the fee which is 1%). In easy terms, Amazon gets paid out $99. The payment processor takes responsibility for the conversion of Bitcoin to fiat, which means that the volatility is only the concern of the payment processor. That goes up for every store accepting Bitcoin through these payment processors. They optionally can choose to go for a certain percentage that they receive in Bitcoin, and the rest in fiat, but that optional feature doesn't see much actual usage due to stores not being interested in holding something that's so volatile.