The customer pays part of that $20, just like he would if it was a flat untipped wage....
$10 cheeseburger + $2 tip = $12 cheeseburger.
That's a 20% increase in price to the customer to get the waiter up from a few dollars an hour to $20.
Get it? 20%. Not 200%. Not 100%.
The thing is, not every part of industry is tipped. Raise the minimum wage, and you raise the cost of living everywhere. It may not happen instantly, but it does happen. When it costs employers more to have workers the cost is pretty much directly translated into higher material costs.
A customer may already be paying for a $12 burger, however, the base price of the burger will rise with an increase of minimum wage.
There are ways in which base prices don't rise (Illegal immigrants, technology advances), however, those are already going to be employed by whatever business that is going to use them.
The other problem with raising the minimum wage is the fact that the primary minimum wage earners work in businesses that have larger impacts on low income earners (Food, construction, etc). The companies that are more unaffected by a higher minimum wage are generally not providing life essential products. The distribution of payment for higher minimum wage is targeted at the people that minimum wage earners use the most.
Doing a government welfare program, on the other hand, has the opposite impact. The burden is distributed across all industries. Not only that, but the high school kid that is living with his parents doesn't rob needed funds from the mother of 4 who is earning minimum wage as well. The funds can be distributed in a more targeted fashion.
Raising the minimum wage is just a bad idea all around. It has little/no impact on its intended recipients.