webmail clients are only blocked 'due to abuse' because it's easy to make bots to register fake email addresses. But even that is not fool-proof. I remember that back when AOL on dial-up was my internet connection, the 'master' account could create multiple 'child' accounts. So even the ISP in that case could allow you to create a 'fake' email for the purpose of registration.
They're also blocked because of a high number of users (proportional to other e-mail services) registering accounts to troll anonymously.
Quite a few registration forms do block webmail addresses for reasons of abuse.
Registration forms that were built in the 90s, agreed.
No, I regularly find modern registration forms that won't accept mailinator or even hotmail addresses (and your service is probably more akin to mailinator than hotmail).