No I have not done any evaluation. It probably depends on the severity of your winters and how deeply the soil freezes. During mild winters I would imagine roots growth does continue uninterrupted even if little or no top growth is observed. I don't know about the walking onions but garlic will break dormancy here when the soil temperature gets to just a few degrees above freezing in late winter. Last year was a very mild winter and I started to observe new top growth in mid February.

My garlic usually does not emerge until spring, but the cloves show roots by December. The Egyptian onions show a little green year round. I stuck some mama Egyptian bulbs in late, and they have pushed up to the surface as they've started to grow.

The mulch is not necessary for garlic hardiness. It's primary purpose is actually to moderate the effects of midwinter and early spring thaws to provide a more uniform ground temp to avoid having the planted cloves from "frost heaving" out of the ground. Garlic roots start to grow very quickly after planting and only actually stop when the ground around them freezes. And the they start again very early as the ground starts to thaw. If you do not get a reliably constant frozen ground the roots will continue to do some growing all winter. Up here in Minnesota we get frozen ground down to around 4 feet deep most winters. Less under heavy snow cover and more in open seasons, even though the more open winters tend to be actually a bit warmer (albiedo) . Garlic root growth here is late in autumn and very early in spring. Some autumns we get some top growth too, but this year I planted very late and got none. I tend not to mulch over winter, and more mulch to conserve soil moisture in the spring and summer.

For me, I always want to cover soil over the winter. Exposed soil is not good. Erosion, weeds etc.

Yes, over-wintered garlic does grow some roots, even just minimal. It responds to temperature really well. I do not think there is any difference of garlic vs, spring bulbs, like Narcissus etc.

I like mulching everything because it keeps away about all of the winter weeds.

I never mulch...mostly because no matter how late in the fall I did mulch, it always ended up with a family of moles or field mice..."living right in the pantry" LOL...they not only had a warm bed, but food all winter, too! We've had some wacky winters here..used to always have lots of snow cover..now we've had green winters, and winters with almost no snow cover, and my garlic and walking onions live just fine through it all.