Off the top of my head, the only ones I can think of are utilities, other banks (so you can make loan payments), and consumer payment processors like PayPal. In the first two, they'll have your ID. In the latter, they do the "make two deposits" thing to verify you own (or at least have access to) the account.
I can't go to Wal-mart and pay via ACH. Do they even take personal checks anymore?
And it certainly used to be common for e-commerce sites to accept "electronic checks" by having you type in the routing and account numbers, with no additional verification. These days, there usually is additional verification.
It is still commonly accepted with no verification for payment for things like utility bills. This appears to be on the assumption that if there is check fraud they probably know the guilty party, because who else would be trying to pay your power bill?
Pretty much anyone with a Stripe account can accept payments via ACH.
https://stripe.com/docs/payments/ach-debit
I worked at an ISP and we definitely appreciated it when our customers paid via ACH instead of credit card. (Less fees.) We were tiny, too; you don't have to be a giant company to get this going.