#Your proposal creates too much disadvantage

The reason that the fog gives advantage to the attacker is that the defender can no longer see the attacker's attempt to shoot him and is thereby less able to dodge the shot if it comes close. Your proposal will create disadvantages where there shouldn't be any, your example case included.

The 5e rules are intentionally simplified for faster, easier play sometimes at the cost of verisimilitude. This is highlighted by one of the other answers here mentioning that in your example if a straw dummy was the defender the shot would be easier which of course it shouldn't be. One might argue that the attacker should always have advantage against the dummy since it can never see the attacker or dodge the shot. That, however, is already accounted for in the dummy's armor class, and that the real effect of the fog is that the defender should lose some or all of his dexterity bonus to AC...but now we're drifting into a more complex rule set...

Other editions of the game better supported different playing styles and you might find that the simplified rules in 5e don't support the level of verisimilitude you'd like at your table. Your table might be better served by AD&D 1st or 2nd.

Your proposal creates too much disadvantage

The reason that the fog gives advantage to the attacker is that the defender can no longer see the attacker's attempt to shoot him and is thereby less able to dodge the shot if it comes close. Your proposal will create disadvantages where there shouldn't be any, your example case included.

The 5e rules are intentionally simplified for faster, easier play sometimes at the cost of verisimilitude. This is highlighted by one of the other answers here mentioning that in your example if a straw dummy was the defender the shot would be easier which of course it shouldn't be. One might argue that the attacker should always have advantage against the dummy since it can never see the attacker or dodge the shot. That, however, is already accounted for in the dummy's armor class, and that the real effect of the fog is that the defender should lose some or all of his dexterity bonus to AC...but now we're drifting into a more complex rule set...

Other editions of the game better supported different playing styles and you might find that the simplified rules in 5e don't support the level of verisimilitude you'd like at your table. Your table might be better served by AD&D 1st or 2nd.