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I did believe I was accounting for spin doctoring by increasing the 0.8% to 20% when I acknowledged different definitions and said "most".

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Your link that you are taking as minimum wage doesn't account for people trivially above minimum wage. Yes, 'most people earn above minimum wage', but that's not the spirit of the discussion - someone earning 25c/hour more than minimum wage is still earning a crap wage. There are significantly more people on a crap wage than "the numbers go up and down a bit from 0.8%".

Similarly, that link only takes the federal wage into account - of course it's going to be abnormally low, given there are states with higher minimums. For example, Washington state has a minimum wage of $9.47/hour. A minimum wage worker in Washington won't show up in a study claiming $7.25 is the minimum wage.

Also, where do you mention 20%?

"Most" means >50%. 70 - 50 = 20

If it's such a small number of people, is there a problem with substantially increasing the minimum wage?

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