In the past, I used curl to get some billing info, add the money to a dedicated virtual prepaid card, then pay the bill, then send an email to a gmail (+paidinvoice) label. These day, at least for my bills, they have pre-approved withdraw directly from the bank. However I guess this is not widely deployed.
If other people did this, but ended up doing it from an insecure machine and lost the credentials / got hacked, I can see why at least some orgs might want to prevent people from doing this. This is a classic over reaction, but a plausible scenario.