This is not a Chabad minhag, it’s a general Ashkenazi Jewish minhag, which only Chabad and maybe a few other groups have kept. It’s an explicit Taz that this is the minhag, and the Taz was long before there were Chabadskers or any other Chassidim. If Neville Chamberlin thinks there’s an issur to spit in a shul, let him explain the Taz. In the Beis Hamikdosh there is such an issur, but not in a shul.
If in a Chabad House you didn’t notice people spitting, look more closely. You will notice the actual Chabadskers spitting, perhaps very discreetly. There’s no need to expel a large and visible glob of whatever, if you swallow before you start Oleinu there won’t be a lot to spit, and you can do it without disgusting people.