I would be very surprised to hear of any European country where retail
prices for non-trade customers are quoted net of tax. ...

That's because VAT is standard across a country. In the US there are thousands
of taxing districts in every state, each with it's own adder to the base rate.

That's not really the case. Some states have lots of tax districts,
e.g. here in New York where every county and many cities set their
own sales tax rate. But some states like Vermont have a single rate
for the whole state, and some like New Hampshire and Delaware have no
sales tax at all. For prices in stores, everything in the store is taxed
at the same rate, but those prices are quoted net of tax, too.

My understanding is that most prices in the US are quoted net of tax
for purely political reasons, that the tax sceptics in the legislature
want everyone to be aware of what the tax rate is, and how much the
state is collecting.

We will leave as an exercise for the student why prices for gasoline
are invariably quoted gross, with all taxes included.

R's,
John