It hasn't hindered the other LCGs all that much.
In Netrunner's case, HB were popular when their big box came out, but that actually had very little to do with the big box expansion and more to do with the state of the game and the ice in the first cycle/core set. Jinteki desperately needed their big box in order to finally become a viable faction, instead of something people tended to splash cards from.
Hell, look at NBN, they haven't even had their big box yet and they are still one of the most powerful factions in the game. The cards they got in the cycles (Jackson howard!) kept them alive and very, very tournament ready.
On the runner side, all 3 have had fairly equal representation, but for a while Anarch stopped being tourney viable due to how better the Shaper and Criminal sides were looking.
More cards does not equal 'better in tournaments'. As I said, people rarely use any of HB's big box cards.
People will pick a faction and keep going with them when the game is young and they keep playing more and more games. New strategies will come to light, but I highly expect L5R to be incredibily varied. Sure you will get flavour of the month factions with great new cards, but the next month another faction will get those goodies.
The only exception to the rule I can think of is when Key and the Gate came out. The winning deck that year used cards primarily from that box, the other half of the Yog Sothoth deck came from various other releases. However, that deck only seemed to work once in the tournament scene, a one trick pony if you will.
tl;dr if the first big box is crab, you may see a larger representation of crab players at gencon that year, but mostly it won't affect the game too much or make crab overpowered in any way.
Edited by Internutt