The first rule doesn't work in practice. I and many others have had the exact same idea, but it doesn't work.
What the first rule essentially says is only the 2nd player is allowed to win the game by pileout. Considering around 30% of games are won that way, you've just replaced a 10% first player advantage with near total second player advantage.
How is that the case? The first player can still win by pileout, they just no longer get the unfair advantage of an extra turn.
Even if pileouts are specifically a problem, the rule could be amended so the extra turn only applies on province depletion.
I think the real problem with that rule is that there's really nothing special about the arrangement of your starting deck compared to the arrangement of your deck after any shuffle...
The starting hand is special: since no decision was made prior to drawing the initial hand, the randomness can be eliminated without reducing strategy in any way.
And continuing down that line of thinking, why not allow players to order their deck whenever they "shuffle"?
Contrarily to the ordering the starting hand, ordering any subsequent hand impacts the strategy of the game.