Haha, Herf..."an important question for sure"
St. Peter's has a public restroom - on the ground level before you go up the steps to the Basilica. No money needed. There is an attendant there (and a guard). There are signs to direct you. Or guards to direct you. They do speak English. They had soap dispensers and hand dryers.
You can always go into a McDonald's and use their restroom - I had to one time - did not buy anything - lines were longgg anyway....just went to the back of the restaurant - used facilities - no money needed - soap and hand dryers worked fine.
As a rule, I "do" carry wet ones and a couple of packs (10 ct) tissue packs daily (out and about.). No need to carry a whole roll. I bring enough packs in my checked to last me the whole two weeks.
eta: The tissue packs can be bought at Dollar Tree or similar stores. They have ones that are 2- or 3-ply - usually come in a 10-pack. If they don't have those, you can still get a 6- or 8-pack. Sometimes, Dollar Tree's store inventory differs depending on demographics, warehouse availability, etc.
One time, at the Trevi, I bought a small gelato. I used their restroom at the back of the gelateria. But, I wanted the gelato anyway, so it was a win-win, lol.
Of course, most sit-down restaurants will have a "patron restroom or two."
Just in case you go somewhere that does require some coin - keep some euro coins on you - usually when you buy something small - break a 5€ if you can - you will get a 1€, 2€ or 50 cent coins.