We use Turnitin via Blackboard Ultra, but am putting this here, as I suspect it'll apply to all VLEs. 
We have a number of staff who prefer to use Turntinuk.com directly for marking, as they can see all their classes at one time. Particularly for those who're Assessment officers etc., and need to validate others' grades etc.
Would it be possible for there to be a link from within the VLE, and remain within the VLE to a "show all my active classes"? 
I think most, though not all of the data would be useful - see image & annotations below. 
The edit option isn't that useful when it's controlled via the VLE (unless the class needs to be extended & that's usually a central role, not one for the graders), equally, delete.  Potentially a dangerous option!
If the class/learning analytics could also have a view similar to the one within a class, then a member of staff could quickly analyse it. 
From an audit point of view, additional data visible in the class list could be: 

• Number submitted/unsubmitted. (Staff sometimes need to chase non-submitters - no need to open a class if they've all submitted). 
• Number marked/unmarked. (Staff sometimes need to chase staff who've not graded ... again, no need if they're all marked) 
• Number feedback seen/unseen.  (Students sometimes comment they've not had feedback... useful to know if they've actually looked!)

That's 3 columns that would be useful in place of the 3 I've deleted in this particular scenario, though I think that would be an enhancement. 

Personally, I'm less in favour of seeing the overall similarity, number of high/low etc. as I think that sends the wrong message that "high = bad / low = good). I think many here are part of the group who tell staff / students - "You have to look at it, you know what you're expecting in that particular exercise" 

Once staff open a single class, then straight to the relevant list of assessments - this time, have "view" to open, but not the editing options.