Oh, I'm sure it has some effect..it has to, actually. Basic thermodynamics, all that lower level warmth doesn't just vanish after a SSW. Agree 100% there.
However, given the cooling observed in the upper troposphere and stratosphere, and the dynamics responsible for delivering those frigid, O^3/H^2O/CO^2 starved airmasses out of the polar upper levels, I'm not sure the effect has been very significant on a high frequency, airmass-to-airmass basis. That's very complicated and each airmass is unique, IMO.
Note how "historically significant" airmasses continue to occur around the globe periodically, despite the warmer background. The severity of those airmasses is determined by chemical processes largely independent of lower tropospheric temperatures. Though they've definitely decreased in frequency overall as the tropospheric circulations themselves are inhibiting the southward propagation of mobile polar high trains.
Right, and if it were as simple as the Arctic being the source, this would be next to impossible, since the Arctic has run so much warmer than the past during the last decade or so.
What's more, the Arctic was actually quite a bit colder in the 1980s and 1990s than it is today, and yet the PNW still struggled to see top tier cold in January over that time frame. Going back to the regional aspect of the conversation.