Yeah ... Rapid Redux would be an example - he's currently won 14 races in a row ...
I think the non-time restricted thing was probably pushed by some trainers who wanted to exploit the NYRA starter conditions.
Hero's Reward just ran 3rd in a Graded Stakes race a few weeks ago - and after he runs here tomorrow - he will have 2+ years of starter conditions in NY even though he hasn't competed for a tag since '06.