What about evergreen shrubs -- are they not propagated this way at all?

Commercial vegetative production uses cuttings or grafting. Layering is impractical on a large scale, requires a branch close to the ground and results in a highly noticeable bend in the new plant. All in all, unless you have one or two plants you'd like to layer, cuttings are a better way to go.

My shrubs that have layered themselves include Clethra alnifolia and Viburnum tomentosum (doublefile). The ones I'd like to do are lilac, Fothergilla major, Korean allspice viburnum, and inkberry (Ilex glabra). Would any of these present problems?

Now, I've got to ask if you are perhaps confusing layering with suckering. The two plants you mention as having rooted themselves into the mulch naturally produce root suckers, especially the clethra. These can be carefully dug, potted and planted out when they're big enough. Layering involves bending a branch (or using one that's already close to the ground), wounding it, pegging it down and covering with a moisture retaining soil mix, and waiting until there are enough roots to allow the new plant to grow on its own. Sometimes plants do naturally layer themselves, but generally not ones that normally grow upright.

I do know the difference between a sucker and a layered branch rooting itself. One of my clethras was mis-shapen when I purchases it (75% off so what the heck), and some of its branches are very low and always get buried in the mulch. One of these rooted, which I discovered when weeding and applying new mulch. Pretty much the same for a doublefile viburnum, which has some very low branches. Both are in wettish areas as well, which I guess must help.

Anyway I will read up about doing cuttings; thanks for the advice.