The blanket policy does have the effects you described, but more _often_ it gets rid of hidden charges or makes it trivial to prosecute them. Americans are apparently so used to these that it doesn't register, but Europe has been pretty effective at ensuring that headline prices must reflect the price you pay so as to be comparable.
Europe wants to ensure that if you say Product X is €10 and your competitor says it's €12 that's not just because you were better at lying than they were, the customers buying from you should pay €2 less for Product X. Transparency is essential to having a working market.
The most common place to see "processing fees" was online where in fact cash isn't an option anyway. A company would offer a product for seemingly less than competitors and then charge a "processing fee" which was really their profit. Committees looking at this stuff found that companies charging processing fees were fiercely resistant to a regulation that made them charge their actual cost, insisting that this couldn't be measured anyway and so the options were:
* Let companies advertise a price and then add processing fees of their choosing. As I said Americans are used to this, but Europeans see it as destroying market transparency.
* Forbid tacking on fees altogether (the option they took) so that prices must end up including the merchant's card processing fees the same way they include customer service call costs, warehousing, taxes or anything else.
* Regulate to try to ensure customers have a real way to avoid any processing fees, likely leading to years of court cases as courts rule out each individual sneaky new trick to make an "optional" processing fee in practice mandatory.
In the end I think the route the US chose ends up being more regressive, but I agree the European option isn't a panacea.