Yes, there are transactions that occur in cash, bank transfer & such. Fail to present a Visa/MC when you rent a car in Dublin or Brussels and you will be in for a surprise. In Dublin, standard practice is to compromise by making you pay a 5X rate... insurance or whatnot. Maybe there is a no-card option somewhere in the city, but it certainly isn't most places.
Yes, you can be an electrician or consultant without accepting cards. No, it is not practically possible to run a store or most customer facing businesses. Even most electricians and consultants accept or at least use cards.
There's no meaningful difference between Europe and the US. This is a powerful and hard to avoid duopoly.