Maybe not immediately, but it is a possibility over time. Nothing is stopping a lighteyed Shardbearer from taking your Shards. There could have been a slow accumulation of lighteyed Shards.
More importantly, however, there are Alethi highprinces without Shards. How did they become or maintain their status as highprinces without Shards? Your argument assumes, at least in part, that no Shardbearer can be ruled by a person without Shards.
I think it is possible that after a period of time, the Alethi conflated the eyes of the Radiants with "lighteyes." Some darkeyed Shardbearers submitted to lighteyed rule; some darkeyed Shardbearers were bested by lighteyed Shardbearers; and, before lighteyes largely refused to wed darkeyes, some darkeyed Shards descended onto lighteyed offspring.
You are assuming that knowledge is perfectly preserved. It isn't—Jasnah's studies are proof enough of that. I would say that one of the more interesting themes of the book is the way that history blurs memory and gives rise to strange beliefs. People could eventually forget that the Radiants had "unnaturally" light eyes once surgebinders disappeared. After a time, the memory of Radiants' eyes could give rise would devolve into a belief that blue- and green-eyed individuals are leaders.