Attacks per round is an abstraction, of course, when using one-minute rounds. There are actually many attacks in one minute -- but we only roll once. If high-level fighters get multiple rolls, it's not because they make more attacks per round, but because they have a chance for more significant attacks per round.
I don't like the AD&D system, but I'm thinking of two other options:
- Keep it as just one roll, but increase the damage: 1+1 dice of damage at level 4, 2 dice at level 8, plus another die of damage every 4 levels.
- Fighters facing opponents of 2 dice or more get an extra attack roll if their level is twice that of their opponent, or two extra attack rolls if it's three times the opponent's level or hit dice.