> Other than that parenthetical, does it work the same way in Europe?
It varies significantly depending on which European country you are talking about (we have massive social variations across our 30 or so countries), and then on factors within each country, down to the policy of specific pubs/restaurants and pub/restaurant chains.
In the UK, I've seen at least two specific cases (I don't eat out very much, even before the pandemic). These are: the waiter hands you a device where you can enter a tip as an amount or just click through to the final payment, OR the waiter themselves enters the final amount - without a tip - and then just hands you the device to press 'OK'. They will add a tip if you ask them in time. I haven't signed a card receipt for years.