Except standard gyros do have to deal with rapidly shifting weights.
There is no PSR if an Awesome loses an arm. Hell there are not even penalties to PSRs for losing an arm.
So it is a rule that makes no sense. With everything else Omnimechs do I don't even see it cutting into the Omnimech's advantages enough to be a big deal.
From a fluff perspective I can see there being a period where standard mechs couldn't because of the lack of appropriate design features as Battle Armor weren't a thing when most mechs were designed or there being doctrinal decisions that standard mechs should not be Battle Armor carriers but that does beg the question why should Omnimechs.
Actually, standard gyros can handle
slowly shifting weights. If one point of damage causes that Awesome to lose the arm, there's no PSR, but in that case the arm was already beat to heck. If it took a shot from an AC-20 to destroy the arm, then yes it does take a PSR (20+ damage).
A point of Clan battle armor ranges from 2 tons (5x 400kg PAL suits) up to 7.5 tons (5x Heavy BA suits) for OmniMechs. An IS battle armor squad would have weights of 1.6 tons to 6 tons. When a 'Mech takes 20+ points of damage it must make a PSR. And 20 points of standard armor is 1.25 tons. Even the lightest (PAL Suits) of battle armor units is heavier than the amount of armor lost to force a PSR.
Now, I doubt anyone ever thought of this when they were writing the rules, but it works out nicely after the fact.