You pose compelling idea!
(Essentially this is how NCAA football conducts their OTs. They give each team repeated cracks from the red zone until the tie is broken.)
A completely different concept was suggested to me by a friend. You might find it interesting. He advocated reducing the number of players on the field for each team by one for each overtime period played.
For example, each team plays with 10 in the 1st OT, 9 in the 2nd, 8 in the 3rd, and 7 in the 4th. After that something else must be done, since 7 is the minimum. (Or they could just keep playing until they drop.) The idea is to open up the game and create more scoring chances.
I believe that the Laws should simply restrict where the players may go on the field during the overtime periods.
In other words legislate a power play situation for each offense.
Method 1:
"At the beginning of extra time a team must designate five field players who may not enter their defensive half of the field and three who may not leave it." Those eight plus two transition players, who may go whereever they wish, and a goalkeeper make up the full 11. This means that once a team establishes possession in the attacking half they have a 7 v. 5 (plus a keeper) advantage. That should tilt the balance enough in favor of the offense that goals will be scored. (The players will need to wear colored shirts or some other method to indicate to the officicals where they can go.)
The penalty for a player who leaves his designated area is that the opponents receive a direct free kick from the top of the arc above the penalty area.
If that system is too complicated, then...
Method 2:
Two 15 minute periods are played to completion. (or 10 or 5 for youth players)
"At the beginning of extra time the team that wins the coin flip shall choose to field either eleven players or nine players. The opponent shall have the kickoff and choice of which direction to attack. The teams shall reverse directions, kickoff, and number of players for the second extra period."
This way each team gets to play one overtime period with a two man advantage. I guess that if they are still tied after that the process could be repeated with a 4 man advantage!
Diving needs to be dealt with harshly, offside has its pluses and minuses, I don't believe that subs should be altered, except for perhaps one additional per team for games that go to extra time, fitness should be a big part of the contest.