I can't imagine Visa would continue to operate in Europe if it wasn't profitable, and I'm not surprised that there are no European competitors considering Visa has the highly profitable U.S. market to buoy their position in Europe.
I cannot speak for other country but VISA bought the Carte Bleue system in France un exchange of following all regulation and not upping the fees.
For two decade before that and since then, all carte bleue are visa and vice versa.
Mastercard exists here as a premium level card giving you tons of bonus, and usually fee with your account. Eg my bank account online is free and almost no fee for anything (I pay 9e for extra phone and wallet insurance and next day replacement), and as long as I keep more than 24k in the various accounts and give me a Gold+ visa card and a mastercard with 1% cashback.
Building a new network from scratch would be extremely expensive with limited chance of success.
Visa can certainly remain profitable even with the fee caps but I find it hard to imagine that they would try to enter the EU market if they had no presence in it and there were already several established competitors there these days.
