LNBF Dish Illumination?

Bear with me. My 12' dish. Buttonhook feed has a Titanium C2W C Band lnbf and a Geosat Pro ku lnbf side car.
I have about a 5 degree offset angle to shoot over the buttonhook mount (ku only). Preliminary aiming of the ku lnbf at dish center showed poor signal because the 'hook casting is about 8"diameter or so.

During heavy snow I had to sweep the dish to get signal. Of course.
Got a little curious and swept only a little circle where the ku seemed to be aimed at. Used my signal meter app on my phone.
Signal increased while I broomed out the snow and when I got a 3' "hole". That's where signal didn't get any higher.
Assumption. Is the Geosat only for smaller dishes a meter or so big?
Yeah?

I've seen the various ku lnbf's with a stepped cone style scalar/feed horn. Forgive me bluddy terminology.
Also although not used in eons. I have a box of old Chaparal Bullseye, Bullseye II feed assemblies.
Can I assume that they are engineered to let the ku lnbf portion see the entire dish diameter?

Also. Scalars still freak me out a bit in their actual function. Oh. Believe me there are many different views of how they work and what they are supposed to do.
I guess when initially setting up my mothballed dish with a new Titanium C band lnbf.
I went by all of the guides and calculations and put it right where it was supposed to be. Signal kind of sucked.
I peed around and peed around with slight dish tweaks until one day.
I moved the scalar back a little bit and peaked the lnbf. Little by little signal strength improved. Possibly 3-5 or so dB.
Reached a point where signal got the best and left it there. So what was the deal?

Under illuminating the dish? "Choking off" the outside diameter?
I say yes. But all of the math and strings and rulers and calculations had the scalar too close.
Also. The "Collapsable drinking cup" type of ku lnbf's. Are they only good for a maximum diameter?