Just want to leave some info here for anyone trying to install FCS 3 Final Cut Pro 7 FCP7 on any OS-X macOS above 10.6.8
THE SHORT ANSWER: (i've concluded) is the FCS installer REQUIRES Apple's Rosetta software (included in 10.6x) or the install will fail with "The Installer encountered an error that caused the installation to fail." HERE is a link that discusses the issues in detail. Also another round HERE i found useful to form my conclusion.
While OSes up to Sierra (10.12) are officially supported by Apple specs – the installation above Snow Leopard is apparently not!
I've concluded how to install FCS 3 onto higher systems (including Lion, Mountain Lion, Mavericks, Yosemite, El Capitan), FCS FCP needs to first be installed under Rosetta (10.6.8) then the newer macOS has to be installed on top of that.
MORE SPECIFIC INFO: I erased a SSD and installed Sierra 10.12x from scratch and tried to run the Retail FCS 3 installer onto Sierra only to fail multiple times and methods from using the original disks to .dmg disk images – it just did't work.
i did the time on this and hope it saves someone some their time