In AE, we do "ride" The Swings, though we might also say "go on" The Swings (or some other ride). I'm not talking about the apparatus at a park where a row of seats are independently suspended from an elevated bar and you provide the power. There, we would "swing" on the swings.
I'm talking about a large, spinning horizontal wheel with seats suspended by chains. The wheel stops, the seats are filled, and then the wheel starts spinning. By my logic, since the riders do not provide the motive power, then they are truly riding.
I'm curious to know if other teachers in non native English speaking countries have encountered the same confusion I have: (Chinese) students want to say they went to an amusement park and "played the ________". They are unaware of this (slightly) unusual use of ride as a noun, and even more confused by the incongruity of ride as both noun and a verb in the same sentence.