Ah yeah, that’s a good idea. This was sticking out to me today as well, how many clicks it is to get to the dashboard.
First the notification…
…which goes to the group inbox…
…which opens a message with a link to the admin dashboard…
…where we find an upgrade is available:
I go through the clicking process, so I make note to archive the message in the group inbox. For large self-hosted sites, this is a way to ensure someone notices an upgrade is available. For a single admin site, it’s a lot of clicking. 
Here’s a proposal: use a notification that links directly to the dashboard as @darkpixlz suggests…
…but keep the admin group PM if no action is taken after some amount of time (two weeks? 
).