Why does the US not add tax to the displayed prices of goods?

Because sales tax rates vary so wildly between state to state, county to county, and even down to the Municipal level.

So, if a business wishes to advertise, it would be close to impossible to list the actual price + tax, when it would vary so much.

Lets say you are a national corporation, you sell your products across the US. And your new product is coming out, with a retail price of $100. If you tried to include the price of sales tax into it to advertise it on national TV, which rates do you use? If Austin, Texas has an 8% sales tax, then the price would be $108, but if over in Dallas, TX, the sales tax is only 6%, then the price would be $106.

Which price would you advertise? If you advertise the 106 price, then the people in Austin will expect 106, not the 108 it actually is for them. If you advertise the 108 price, then the people in Dallas would be expecting it to cost more then it is for them, and some might decide not to buy it.

So, to account for the varying tax rates, they just advertise and list the retail price, so everyone knows they are paying the same price for the same product, and the only difference is the sales tax, which people are aware of in their communities.