Dave has a point.... maybe not so much of a band aid.
What I noticed was that when i went through every component and took all teh slop out o it the situation got better, but it never was resolved. that stabilizer made it so the wobble wouldn't resonate which "fixed" it.
Maybe the reason it didn't do it before is because everything was "perfect" now that it's not it can start then quickly runs out of control. maybe add one and just see? I wouldn't say it was worth my effort to rebuild everything only to find that 99% of the crap i went through was perfectly servicable and within specified wear limits.
They did come with steering stabilizers from the factory..