littlerack web interface may solve it
If I install a web client somewhere, I still need to make the very same connections a mobile client would do...
The only difference will be what device is starting the connections.
If a company offers their own implementation of a web client, they are still making the very same connections to their servers.
The fact that the owner of the web client is the same as the owner of the mail service, doesn't prevent them to make the very same connections.
secrec The point is that its not even an application that does anything USEFUL
It's giving you the notifications, which is a desireable feature to some people.
Their use case is stated clearly at the very first line of their readme.
If you can't find this app of any use, it doesn't mean it's useless, just that you don't need it.