The comparison to credit card theft is inappropriate; you are not exposing your private information in the same way you are with a credit card. When you pay for something online, you are transmitting your credit card number over the wire. When you pay for something with BTC, you are sending merely a signed transaction over the wire. When submitting a credit card or debit card number, you are trusting that the merchant or people that work for the merchant will not steal all the funds from your account, and that is why you need extra guarantees from your bank (granted, those guarantees are not always upheld
https://nypost.com/2017/12/14/after-losing-familys-846k-inhe...). With BTC, it is not a requirement for an online transaction to expose credentials which allow an untrustworthy actor to drain your account. Similarly, you do not have as much risk from a Man-In-The-Middle attack with BTC.
For sure there are some untrustworthy exchanges as there are untrustworthy ICOs, but you have two options that are far better. You can rely on US Banking regulations and pay higher fees through Coinbase and Gemini, where your deposits are insured; or you can use a decentralized exchange like etherdelta that relies on atomic swap contracts.
As you mentioned, Monero (and potentially z-cash) are there as solutions to anonymity.