Bitcoin is backed by processing power
It is not correct to say that Bitcoin is "backed by" processing power. A currency being "backed" means that it is pegged to something else via a central party at a certain exchange rate yet you cannot exchange bitcoins for the computing power that was used to create them. Bitcoin is in this sense not backed by anything. It is a currency in its own right. Just as gold is not backed by anything, the same applies to Bitcoin.
Bitcoin mining and gold mining aren't the same. When you mine gold, once the job is done, its done. You have gold and can sell your equipment because it no longer serves a purpose. Bitcoin miners continue to process transactions, so they support the existing network and allow you to validate transactions not only for new bitcoins, but also those who are 10 years old.
If we would compare this to gold mining, the machines used to mine gold would also be the ones used to clean, cut, melt, polish, weigh and count your gold.
If miners were to stop confirming transactions we'd have a drop in bitcoin's value. If the same thing were to happen with gold I'd expect the opposite effect.