It's not in question that Assassins don't auto-get Independent Action.

Those that have Independent Action are explicitly specified. And they are:
King Hassan (no comment, it's Gramps, he's broken)
EMIYA (Assassin) -- because it's Kiritsugu
Hassan of the Serenity -- because she had it in Prototype
Ryougi Shiki -- actually a human

It wouldn't need to be specified if it was implicit.