That may be the worst idea ever.
1. You know how exhausted everyone looks after just 15 minutes ET right? Professional soccer players cover a lot more ground than little kids. They would be too tired to move after just 2-3 ETs. So the game would be stagnant and awful to watch, and there would be little chance of goals being scored, which would mean... more ETs?
2. Removing players after each ET would mean even more running for each player to attack / defend, making them even more tired than if it was 11 v 11. And it would get worse with each ET.
3. In your local crappy youth whatever, this may have led to attacking play because who cares, but in a situation that actually matters with trophies and money and jobs on the line, it will lead to the attacking players being subbed off first. Eventually it would be a bunch of defenders and defensive midfielders. All the Feilhaber and Fagundez types would be taken off as superfluous luxuries. Eventually maybe each team would keep one attacker, either a speed guy if he had any legs left (unlikely) or a Conor Casey type they could boot the ball up to and hope to get lucky. Exciting stuff.
Like I said regarding the away goals rule, in soccer the consequences are often the exact opposite of what is intended.
Did the golden goal lead to attacking soccer? All you have to do is score and you win, seems pretty obvious right? Teams clamped down on defense, to make sure they didn't give up the goal, and it was far more boring than playing for a fixed period of time.