This is useful to show average site owners that wouldn't know about pagespeed in teh first place. If they see green numbers they feel good or notice a sudden slowdown.
+1
As a developer I never use a hosting provider's pagespeed insights as they are usually basic and delayed. As I am optimizing I use better tools for this in another window/screen. I doubt this would be used by developers except at the end to show the client.
With this in mind, I think it should show a history of scores. It should also be hidable by the admin as some sites will never get nice numbers despite being fast. Some clients get hung up on this because of incorrect blog articles telling them they need high 90's in everything. If they never saw the red numbers in the first place they wouldn't have searched online and gotten all worked up.
Should also send alerts if pagespeed insight metrics worsen. Ie load time change from 4 to 10 seconds should send an email.
