The only time I ever use a CC is when I'm dealing with an American company- because it's either CC or PayPal with them.
Visa and MasterCard debit should work anywhere online regardless of origin.
These cards don't have a 16 digit card number and CVV that you can in online shops and pay with those. This has the advantage that these numbers also cannot be "stolen" and then used by criminals to buy things.
Debit cards are really just for physical payment, where you either use contactless payment or by using a PIN. For online shopping, the shops talk to the banks by using an OAuth-like API that can be used to either pay once or set up a monthly payment. There's a few different of these APIs throughout Europe, in each country usually only 1 or 2 are used. Some examples of these APIs are: Bancontact, iDeal, Giropay and Sofort
European webshops usually only support the ones in the country that they are located in, and for all others they either support credit card or bank transfers (which are free and standardized within Europe). American webshops usually only support credit cards, so that's the most common reason for people to have one.
Dutch here. We don't have points or cashback. Purchases are insured with standard bank debit payments as well, as are chargebacks. I do own a credit card but that is purely for the occasional (usually US) webshop that doesn't accept European debit cards, or as a backup payment method on vacation (my debit card once broke on vacation and my family had to send me money through Western Union)