Any Ascended character is a (comparatively) senior member of the Inquisition. They have served as Throne Agents, which in of itself confers considerable authority, and have received training and authority beyond that. They are exactly the kind of guys who would deal with a Grey Knight. The Inquisition doesn't sweat that one of their senior agents comes from the gutter before he served in the Inquisition. If there's a daemonic incursion and he's one of the best men on point he will be and if that means working with a Grey Knight why would they have a problem? The Inquisition has agents from all kinds of backgrounds. The relevant part is if he can do the job and as an Ascended character the answers tend to be "yes".
Now here's this crucial line of yours:
I just don't think that one should strictly follow the fluff at the expense of what might be fun
Now there's nothing wrong with the statement or the sentiment behind it, but it is naive for you to expect FFG to do that. Every book they write goes through GW readers to make sure it is consistent and true to the background material and their liscense obligates them to do so. They aren't ever going to produce material on Grey Knights light because of it. There's nothing wrong in you wanting it, but you or another fan who feels the same way is going to have to do the work.
And yes, my opinion on the model is my opinion. O_o What else?