You can get Gentoo up and running pretty quickly by following the handbook. From memory it’s easy to miss one or two clear instructions because the styling of the handbook can add more eye-catching weight to the explanation than the actual commands. So be sure to re-read areas where things don’t seem to working out.
Gentoo also has a binary repo if you don’t plan to stray from whatever installation profile defaults you start off with.
I can’t confirm a simple server install of Gentoo is somehow more lean than any other distribution.
I’ve used gentoo-install with success previously although I don’t know how up to date it is.