>>and the heating systems that warm our house also dry the air indoors.

Trivia: it's not so much the heating system drying the air as that as the temps go down outdoors the absolute amount of moisture the air can carry goes down as the air gets colder; warming the air just raises the amount of moisture the air could theoretically hold. Relative humidity goes down when that air is heated because it's a measure of the relation (hence "Relative") between actual moisture content and theoretical max has changed - because the max has gone up. If you look at dew point, it's pretty much the same between indoor and outdoor (excepting indoors the moisture we exhale gets concentrated).

All that said, shivering seems to have a positive effect as it prompts secretion of succinate, which in turn stimulates mitogenesis (more mitochondria). The research I've seen examined fat cells, where more mitochondria can turn white fat into 'beige' - but they weren't examining, say muscle or nerve cells, so it might be a systemic effect and it might not. I can think of reasons it could be either, but biology doesn't work in a 'logical' way.