Woocommerce? It works with BitPay. It won't give you bitcoin though, but it will give you fiat. Who wants that? Bitcoin is something you hold, not something you take and then flip away. Simply export your site onto a proper hoster, and then use the blockchain.info API.
In last 24 hours Bitcoin's price nearly drops 4%. This is what all of the accepting bitcoin payment merchants wouldn't want to see that. They want the stable fiat money. So they rely on the payment processors.
True, but when looking closer to those merchant system, i assume the only way to make it profitable is to sell those coins instantly or once a day.
Bitpay for example have a revenue of millions, if they would be depending on a stable price, they would be broke by now.
My guess is that use exchange addresses or have a sufficient amount of money on the exchange, to sell when they handle a transaction for one of their clients.