In my experience, some banks offer ACH pushing through Bill Pay but often an ACH is only made if the recipient is on a master list of known institutions (like for paying your electricity) and if the recipient is not on this list then a check is sent by mail. YMMV though

It is, but assuming that GP is from Europe, the confusion will be around the distinction between bill payment and P2P payments, which seems arbitrary to bank customers in many other countries:

With SEPA credit transfer (and other similar systems), bank customers can send mony to any other account directly and automatically by only stating the recipient's account number and amount. There is no list of well-known/trusted(?) payees; businesses and individuals can receive money from other businesses and individuals alike.

In the US, for reasons that I don't fully understand yet, this type of P2P push ACH payment is generally only offered between accounts that have been verified to belong to the same person. I suspect that it's largely a question of money transmission licenses, the recipient's unwillingness to freely hand over their account number (due to concerns about unauthorized ACH pulls!) and/or fraud liability.

Bill pay on the backside is ACH push but users never see it. Bill pay will send physical checks as well.

ACH push isn’t that common at the consumer level. Very common for other things like payments/payroll.