The EU isn't one homogenous place in terms of local habits and practices.

In at least some countries you can totally get away with being cash only (although less so since the pandemic).

Ok I’m not sure if it’s an EU thing or not then but I think here in Italy, if you run a business, you’re not legally allowed to go cash only.

It’s just not an option. You must accept traceable payments either via cards or direct bank transfer.

To provide another data point, in France you are allowed to refuse credit cards or checks. Cash is the only payment method that you cannot refuse but there are exceptions for that as well.