The pharmacy that I use has a very busy and probably very expensive web site, but it has bugs, and any time you get off the happy path, you have to call. For example, when I get a prescription delivered for my brother-in-law (who has disabilities and lives with us) the pharmacy only gives the name on the credit card to the delivery driver (this is a known bug, for over a year) so the driver shows up and asks for the wrong name. Usually the pharmacist figures this out and gives them the right prescription, but if they don't, the prescription goes into a black hole. It can't be picked up at the pharmacy, it can't be scheduled for delivery again, nothing can happen until I call and get it fixed.

Helping my mother set up online access to her pension, the only way they will grant her access is if they call her at a certain number and she answers the phone.

I can't think of any other examples off the top of my head, but I despise talking on the phone and will go to great lengths to avoid it, and yet I have to do it maddeningly often.