It still would be possible to link UU / UB possesion to several conditions (being the first to research one/two technologies, plus having a resource or type of land in your empire, plus having built certain building)... maybe relaxed for the original owners of the UU / UB.
This would include additional "races" in the game, and risk some civilizations getting and edge over others.
But it could be compensated by allowing other players to research in less time once the UU/UB is assigned (i.e. you get Legions if you are the first to reach Iron Working and build a Swordsman in a city with Barracks) - yet after you get assigned the Legion, iron working and/or swordsman cost is reduced by 10-20% (which is the plus you paid to have the UU/UB). Rome could start with these discounts already applied, so it is easy for them to get the Legion.
Very interesting idea! It could certainly work, but won't the civ with the tech advantage also have the most UB+UU. Or, will there be a max per civ? In which case, which idiot is ever going to get modern UU?
Nonetheless, I really like the idea.