I never thought about paying staff via tips being a tax dodge for the restaurant if it lets them avoid raising prices and thus taxable revenue.

If the price increase is actually linked to higher wages for the staff, there should be no increase in taxable revenue for the restaurant, because the wages are deductible expenses. But there is still the matter of sales tax (paid by the customer), and the tax payable by the employee on their wages (not sure if tips are technically taxable income already, but regardless, you can bet most staff aren't paying taxes on them).

The BBC article's "sales tax" reason sounds good - unless you know that several US states do not have sales tax.

The actual history of tipping: https://en.wikipedia.org/wiki/Tipping#History

And a bit on Sales Taxes: https://en.wikipedia.org/wiki/Sales_tax#In_the_United_States

From old family stories, I figure the real reason for tipping service workers is that both employers and customers want there to be plenty of service workers available - but the demand for their labor (restaurant customers, whatever) is very unpredictable compared to factory, farm, or gov't work. And service industries can't afford to keep a lot of idle workers around at full pay. So tipping is a simple revenue-sharing scheme, wrapped in social niceties, that boosts/cuts the service workers' income based on whether business is currently fast or slow.