We had more troubles with work with "the cloud".
Setting it up was just pressing publish in VS + git and 1 time the domain+site ( 5 minutes in total ). So I spend less time on it.
Everyone seems to forget that every deployment needs to be configured somewhere. If not by you, someone else.
The server is windows server with IIS. Nothing special, you can get it for the price of an 1 Shared App Service in Azure ( 10 € / m = 1 Windows Server) vs 8€ / month ( 1 Shared App Service).
I didn't had to do maintenance in the last year. But I did create a custom statuspage. But it's used mostly to deliver updates of clients when I deploy ( it creates a RSS entry when I deploy, which clients can receive).
Clients are notified 100% of the time with updates from the developer as soon as it happens, if they want it :)