Additionally until recently most political parties and people in the EU didn't see this as national security related infrastructure. That's why it was allowed to be privatized and handled by external companies. There is a lot more critical digital infrastructure that is being moved away from. Think Microsoft office suite, operating systems, cloud systems and more.
Most importantly: These cards give customers fraud protection, which is in many cases essential when making online purchases or when traveling. Which leads to more sales for sellers as well, when that worry is removed from the shoulders of customers.
Visa and MasterCard are everywhere though
And regarding Visa and Mastercard being US payment networks: This is now undoubtedly the case, but for much of their history, their EU subsidiaries were actually independent cooperatives owned by large European network member banks. They only sold their respective stakes to the US parent organizations in the late 90s, back when European banks, in a pretty severe blunder, considered cards a thing of the past of no importance for the future.