No, that washer is not essential, it just makes it a bit easier to fit the headset wrenches together to tighten the locknut to the top race. In fact, many of the washers do not have the internal tab so they don't act to keep the locknut from wanting to turn the race as it is tightened. The steering should be smooth and drag free right from the start. It sound like the adjustment is too tight.

I think I'd find a different LBS for any future work.