While we're waiting for an answer for why pay toilets are common in Europe, here's an answer for why they are not common in the United States:
In the past, pay toilets were as common in the United States as they are today in Europe, and the push to remove pay toilets from the United States can largely be attributed to the Committee to End Pay Toilets In America, or the CEPTIA for short. The CEPTIA was founded in 1970 by a group of college students to end pay toilets. There main arguments were that 1. Fee-free toilets are a human right, and 2. Urinals could be used without charge, but stalls were paid, meaning that women were put at a disadvantage. Nik-O-Lok, the primary manufacturer of pay toilets in America at the time, argued that pay toilets discourages drug users, muggers, and vandals, and that their revenue could be spent to improve the toilet experience. Those who owned pay toilets also made some good money- $50-200 per toilet a month, or $225-900 adjusted for inflation. Finally, while this doesn't seem to have been a major argument at the time, free toilets discourages public defecation and urination.
Public opinion sided against Nik-O-Lok and pay toilets because people simply don't like paying to go to the bathroom. Politicians sided with the public because this was an easy issue to score positive PR, and in 1973 Chicago voted to ban pay toilets. I don't know the exact timeline of the decline, CEPTIA disbanded in 1976 on reasons that its mission was mostly achieved. While there are some faint rumblings to bring back pay toilets in the United States, nearly half a century without them has made an America which believes that pay toilets are unusual and unneeded.
In summary:
America used to have pay toilets, but a campaign led by the CEPTIA led to their removal.
The CEPTIA was non-combative, and general public opinion was against pay toilets.
There was very little support for pay toilets aside from those who made money off of them.
Legislation against pay toilets was an easy way to garner good will from people who use restrooms (i.e. everyone) and from feminists who wanted equal access in public toilets.
Links:
Why Don't We Have Pay Toilets in America?. Primary source for this comment.
Ban on pay toilets disputed. News article from 1974 on the issue.
Group Seeks to End Pay Toilets. News article from 1976 on the issue.
Pay Toilets Are Illegal in Much of the U.S. They Shouldn't Be. A fairly recent article advocating for pay toilets.
Edit: Pay toilet movement attacks capitalism. A sarcastic 1975 article against pay toilets.