Do you mean only one chance (roll) at fumbling per round, or you mean you cannot double fumble... because of it is the latter then it is hardly any different as many fumble results will disrupt all the other attacks as well.
Only once per combat round. Regardless of how many attacks a character has he can't fumble more than once in a round.
Critical and Fumbles are not about being balanced based on level they are about being Cinematic.
So if he rolls 2, 4, 1, and 20 then he doesn't fumble?
As comparison if he had one only attack he would have rolled just 2.
If he rolled that he has a chance of a single fumble. If he had rolled 2, 4, 1, and 1, then he can't fumble the second natural 1 because he only has one chance of a fumble per combat round.
So you protected against an insanely bad string of luck, but the issue of "chance of fumbling increases as the character gains more attacks" remains true. As long as more than one attack qualifies to potentially result in a fumble, having more attacks will mean you will fumble more often.
Critical and Fumbles are not about being balanced based on level they are about being Cinematic.
I also want to add that I've removed the crit confirmation from my Pathfinder game as it's a real killjoy. If you roll the crit, you get the crit.