They can do that but that costs money so they never will. We worship the dollar like its a god above gods. No way we're going to willingly do something that costs the rich a dime more than absolutely necessary.
VAT is deductible if the buyer is a VAT registered business. The amount of VAT is still printed on the receipt by law, so you can still know how much you pay.
Companies already manage seperate ads in the EU for language, currency and price discrimination reasons, it's really not a huge bother.
So customers cannot deduct VAT from their personal taxes...
I also doubt that separate ads per township are common in EU. For example, in Los Angeles the city is made from several separate townships with different sales taxes, two shops on the same street may have different taxes because one can be in LA and another in Santa Monica.