Mechs have very, very articulated hips. There's instances in the background of Dire Wolves (Daishis to the Sphere) losing a leg and managing to not only not fall over, but balance itself well enough to finish off its attacker with a torso mounted RAC/5. Course, that particular one was Victor Steiner-Davion, AKA God's Gift to Battlemechs, but it's still quite possible for mechs to remain balanced on one leg.
The movement we've seen in the past in Mechwarrior games is not at all representative of mech mobility. Mechs have muscles, the myomar on a humanoid mech is almost identical to your own musculature system, and the mech, as a result, moves like a person. Mechs can do Gymnastics if they have to. (examples: Decision at Thunder Rift features both Shadow Hawks and Stingers doing shoulder rolls, and Falcon Guard features and entire mech Trinary, from Fire Moths up to Dire Wolves, doing Jumping Jacks)
Now, all that having been said, the way it's handled in the boardgame IIRC, loss of a leg can happen one of two ways, if it's simply damaged until it stops working you end up like in MW4, you have to make a pilot check to keep from falling over at that point, but if you do you can then stand up and move at significant reduced penalty (1hex/turn, about 10km/h). You can also have the leg completely severed by a critical hit. In that case you fall over automatically. You can then crawl at 1hex/turn. You can also stand up from the loss of a leg, but can't move except via jump jets. Good luck staying on your foot when you land though.
When you've fallen in Battletech you're also not completely hopeless. You can still fire your weapons, though there are limits. (Both arms weapons, or one arm's weapons and torso weapons - using the second arm to prop yourself up) And when being attacked, if your attacker is more than 1 hex away you are actually harder to hit, though your a sitting duck if someone does get 1 hex away from you.
If they could implement some of those things (imagine going prone with a Gauss rifle for a more concealed and stable sniping position? Or a greivously damaged one legged mech doing a suicide DFA on an enemy?

) It would strongly improve the immersion of the game, IMHO.