I think the real reason it's different in the US versus Europe is that modern banking developed independently in each place before travel and communication between them became as cheap and easy as it is now. Each place has a standard banking service for paying money to people, which is easy and convenient and free of unnecessary fees, which just about everyone uses. In the US, that's checking; in Europe, it's bank transfers. In each place the *other* service is not standard and therefore is impractical and a hassle because of fees and general unfamiliarity and lack of standardization and mostly because it's not the standard way of doing things.
I'm afraid this analysis is misplaced - at least for the UK, I'm not so sure about the guys on the continent. It misses the fact that as recently as my childhood (that is, about 20 years ago) our system was your system.
I remember my parents used checks (well, cheques) extensively/exclusively - "bank transfers" were almost unheard of as far as I remember, and cheques were noted as having the advantages you describe. When I was a teenager and got my first bank account, I remember trying to make a direct bank transfer and finding it (inexplicably) inconvenient, expensive. Even when went to uni, I remember paying most stuff like rent and tuition by cheque, online banking was still barely on the radar.
In the last 10 years online and mobile banking has exploded, making easy and free personal transfers commonplace, and that's changed things a lot. Now I can't remember the last time I wrote a cheque, probably about 5 years ago, most of my friends would say the same.
But they haven't vanished altogether: old people probably still use them predominantly, anyone my age or above will be well familiar with them, all sorts of people doubtless still use them, but in overall trends I think they're on their way out. And not from "unfamiliarity", or having it forced on us, we're just giving up on them out of choice, preferring the alternatives.
Recently there are newer services that have been introduced (since the advent of inexpensive internet access) called "direct deposit" and "direct withdrawal", which are probably somewhat more similar to how bank transfers work in Europe, but they're intended for things that happen on a regular basis (like, every month)
That sounds more like what we call Standing Order or Direct Debit, which isn't really the same thing as the general "p2p" bank transfers which people talk of displacing cheques - although it certainly has done that for many uses such as rent, utilities, telephone etc.