Earlier quoted context omitted.
Collecting credentials and reading mail from server is standard practice across hosts of email clients, since it’s the only way to get push notification working. If you don’t trust their server you probably shouldn’t use their client anyway.
Apps can't just send notifications? If I had a mail client installed, couldn't the app just periodically connect to my mail server using my internet connection, see new mail was available, and then pop up a notification so I'd know? That seems much more secure than having a third party collecting my passwords so they can connect to my mail server from their network using my password just to see if any new mail is the…
They can but consider this: How would they know to send them?
If you are adding a provider only locally there's no way for the app to know to generate a notification if it doesn't have an IMAP idle connection open constantly or scheduled polling. If the app is shipped by the email provider (like gmail app for ios using a gmail account) they can send pushes without leaving connection open or polling.
The point of having your details with the app developer is that their servers will remain always connected to your email provierr and generate pushes that will go over Apple or Google's push network.