No, servers in the US do not make $3/hr; they're required to make minimum wage and have a starting backstop wage (that's the $3 you hear about), and make as much more than minimum as their tips allow.
Of course, the US minimum wage is much too low to live on in most urban areas, so the minimum wage part is cold comfort. But servers in $$/$$$/$$$$ restaurants make substantially more than minimum wage in tips. By OpenTable's scale, a $$$ is ~$40/customer; a 4-top would be ~$32 in tips, a normal table is turned in 1.5 hours or so, and you serve multiple tables simultaneously. You can work out a rough range of outcomes in your head. Service staff come in an hour or so early and may stay a bit later to close out, and that brings the average down, as does sharing, splitting, or pooling, where that's allowed to happen, but it's still much higher than minimum wage.
Obviously, this applies to higher-end restaurants. But then, higher-end restaurants are where the "no tipping" movement is happening.