Yes, is totally reliable!
I've been selected this year for a Fulbright Scholarship in IR/Political Science for Italy and my application to schools will be managed directly by the IIE on my behalf.
There are, of course, pros and cons. Regarding the pros, well, you don't have to pay fees and everything related to your application is guaranteed by the IIE which, in case there is something to amend, will contact you.
The cons...you don't know anything about the status of your application until the IIE won't inform you and, moreover, the IIE has a certain margin of discretion to decide which universities are suitable for your profile.
I think the latter is the worst element of the Fulbright scholarship.
I hope to have helped you!
Stefania