Why would anyone want this?
Being forced to level abilities on a fixed, linear progression strips a champ of a degree of strategy or versatility, to say nothing of practically crippling it right out of the fountain as you are stuck with only a single basic ability while everyone else has all three abilities and their ultimate.
And losing levels and XP on death while your enemy still levels like normal? As if being underfed weren't bad enough, now your level gap can grow exponentially as everyone else continues to level up and keep their levels while yours continue to diminish, making your odds of recovery a little over impossible.
And what, again, is the payoff for accepting this system? A champion with inherently stronger base stats than most if not all champs their level, and/or can level up much more quickly than the rest? Well guess what? Balancing that is going to be a nightmare because either they level up too fast early-game and quickly snowball out of control as he curb-stomps the enemy and leads his team to victory while his opponents don't even have the time to farm, or someone can kill them first and they revert back to level 1 and have to start all over again while their opponent keeps all their levels and keeps gaining more even after they finally die, so now the tables are turned and this guy can barely retake level 5 before the game ends.
It's like you wrote a system for an entirely different game and then tried to arbitrarily shoehorn it into League of Legends without any consideration for game balance or all the champs that don't work the way he does, which is to say all of them.
I will say though, a compromise I can picture would be a champ that has a passive that gains stacks as he makes kills and assists similar to but independent of his XP gains and levels, which power up his abilities in sequence independent of normal level purchases, and are lost entirely or partially on death. Kinda like Viegar or Nasus except they make him more ungodly powerful the longer he keeps his killstreak up. Unfortunately even this alternative system could bring its own few cans of worms to the table depending on how it's implemented, but it would keep him more viable under more than just ideal circumstances like the original system would.
