Hi Mark,
All of the shopping malls have free, public toilets, although I've heard that WestEnd is going to start charging (card accepted). I'm fairly confident saying that the hot air balloon park and Olympic park have free public toilets and other parks are moving in that direction. Great Market Hall has paid toilets on the top floor and in the basement across from Aldi. This is when it's nice to have a bit of cash/coin.
The nice thing about European restaurant culture is not being rushed. If you stop in a restaurant for a meal or a cafe for coffee and cake (My faves: Művész Kávéház near the Opera, Ruszwurm in Castle Hill, Auguszt Cukrászda with three locations) you can linger at your table without feeling rushed. You'll have access to the restroom without a fee.
Some fast/quick food places, such as Starbucks or McDonalds, will require a purchase and will provide a code on the receipt to open the restroom.
Many hotels are starting to 'crack down' on non-guests attempting to use restrooms. For example, the Hilton in Buda requires card access to the restrooms.
Tap to pay is the common method and Visa and MC are almost universally accepted. I don't always have as much luck with my AmeEx.
Legenda is the standard for sightseeing river cruises.
TTM shared my favorite: Pörc & Prézli, a sweet little restaurant behind the basilica. Another location that is really, REALLY Hungarian is Nagyi Kifőzdéje (basically Grandma's Kitchen) in Buda. You may need cash and English may or may not be understood, but the food is so good. And it really is like sitting in Grandma's kitchen.