Can you post some recording where this alleged flak rush is being used successfully in MP? I can imagine flak spam working to some extent later in the game, but a flak rush exactly as you describe it starting with "drop down two military lab at the start of the game" would be astounding to see working vs. another skilled human [TEC/Advent] that spammed LFs and rushed with them too.
I've seen "pro" LF spammers just sit back for a long time building up their force (to 50-80 ships) without initiating any serious hostile action in enemy territory in the mean time, but that's really a bad strategy; someone like Grimm or Doci would laugh at that kind of "LF turtling". Since you watched and commented on the 1v1 game they posted, you probably know why that's so. My guess is that if you do run into someone who spams LFs but just turtles/defends with them, flak rush might work because it would be then equivalent to a [later] spam if the enemy doesn't press you early on.
The other thing that might make this possible is starting at the max distance (between HWs) in those 10-slot random maps, which is 5 jumps (perhaps 6 jumps is possible too, but I've not seen it these maps) as opposed to the "close quarters" 3 jumps that seems to me is the common setting in those maps (occasionally just 2 jumps shows up too, but that's also rare). 5 jumps separating HWs is more common in the 8-slot random maps (compared to 10-slot ones) and perhaps becomes the prevalent distance once you go to 6-slot maps (where 6 jumps separation can also be found occasionally), but I'm not sure how often those 4v4s and 3v3s are played on ICO.
So, recordings or at least some more details are needed to see the full picture that you're painting...