Red Dots / Holo sights work different than traditional scopes. There's no eye relief associated with them. There's no traditional Field of View on these type of optics. On my one 16" AR, I have a 3X-9X 40mm as my primary optic and an AimPoint T1 micro redot on a 45 degree mount forward on at the end of my rail. The T1 is about the same size as that little FastFire you got. You're sort of looking through to optic to the target. The the perimeter of the optic disappears and you just really see the dot floating. You don't need a large diameter houseing for a red dot. Actually smaller is better. If anyone talk Field of View or stuff like that with a Red Dot, simply dismiss what they're saying. They don't know what they're talking about. The T1 is one of the most popular red dots with the military (who have the option to choose), 3 gunners, and other tactical folks.
Don't worry about "viewing screen." Seriously there is no such thing really when you're talking red dots. That being said, the FastFire III is a fine little sight. It's not bullet proof like the Aimpoints or Trijicons, but I'd still honestly take it over an Eotech any day... but I really don't like Eotechs at all and would never put one on any rifle...your mileage my vary however.