In non tipping cultures, service in restaurants is generally poor.
While I do think the wage should go up slightly, you do understand that restaurants operate on very very low profit margins, so any increase to the minimum wage really hurts low profit margin businesses, such as restaurants.
My daughter works at a Utah county restaurant and averages $11-18 per hour. That is a pretty wide variance, but she's been able to make enough to put herself through BYU.
If you raise the wage up too high, labor costs will increase and profit margins will drop leading to more restaurants closing, or it will probably result in the end of tipping and result in lower wages for servers.
It's not as simple as you postulate