There is no "problem" to be fixed. League-exclusive uniques serve the exact purpose they were designed for: to reward players for playing on the current temp leagues. Ever heard of collector's edition products, or limited-production items? Because they exist in other games and in real life too!
1) Lamborghini Reventón: Only 21 units were ever produced. Artificially limited by the Lamborghini company, and all units were sold out before the very first unit was even manufactured.
2) 1st Edition Charizard: Printed only in 1999. Artificially limited by the Pokemon company. The card is now worth upwards of $2000 USD.
3) Black Lotus: One of the "Power Nine" cards in Magic The Gathering. Printed only between 1993-1994. Still one of the most powerful cards in existence. Artificially limited by the Wizards of the Coast company.
Do you see anyone complaining that they weren't born early enough to purchase a Lamborghini Reventon, 1st edition Charizard, or Black Lotus card? No. So stop complaining about league-exclusive uniques in PoE. Early adopters will always be rewarded in nearly every aspect of life.
(All these real-life examples don't really translate within the structure of this game but...)
The first two examples you gave are just vanity items that serve no purpose other than being rare. The Black Lotus card was created early in the alpha stages of the game when it wasn't really known how to balance the game and much like many other cards created in that time it is now considered severely overpowered. Black Lotus along with a large number of those early cards have since been banned from official tournaments.
My point is that while these items were only available for a limited time they don't provide any actual competitive advantages. They are similar to the alternate art items from race rewards and limited-time microtransaction items in the shop which I have already explained are fine since they don't affect gameplay.
Uniques from temporary leagues, however, are not like these items because they do affect gameplay and give advantages to players. This system creates an extra needless gap between veteran and beginner players and only discourages the arrival of new players. No quality game should ever give permanent gameplay advantages to players just for being around at a specific time.
Also, I know that PvP is practically nonexistent at the moment but there may be a time in the future when it becomes a prominent feature in the game, and when it does exclusive character builds will be an obvious issue.
Edit: I just want to reiterate that I completely agree with the notion of rewarding players for participating in one-time events but these rewards should be in the form of cosmetic items or useful items that are eventually made permanently available.