Yes, private shipping is perfectly fine. This is not a shipping embargo. USPS is just not taking packages. Letters and flat rate will continue to be processed.
(declaimer: I am from Hong Kong, and have worked on some import tax/declaration system)
Import tax and tariff relies on clear and accurate import declaration.
For DHL/FexEx imported package, you can fine them when the declaration isn't right.
For USPS imported package, you can't fine USPS -- they didn't know what's in package when it arrive. You can't fine Hong Kong Post or China Post, because they are not US entities. Rejecting parcel in bulk is one of the reasonable option if they can't get the shipper fill in the correct declaration form.
If the basis of this move is that the USPS haven't figured out how to collect the tariffs on imports from China you would expect other couriers to be implementing the same measures very soon.