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Because minimum-wage type labor demand is inelastic relative to the actual wage, e.g. regardless of the minimum wage, McDonalds needs at least a few employees on staff at all times - there's a point where they can't cut labor any more, but their revenue is still high enough that the increase in wage payments wont put them out of business, just lower profits.

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I agree with that. However, nothing is perfectly inelastic, and so increasing the minimum wage would still increase unemployment, even if only by a little amount.

Then let's assume it's just highly inelastic. If raising the minimum wage causes a small decrease in employment, but the raise is high enough that there is still an increase in the total wages for all workers in the economy, that puts more money in the hands of consumers, which potentially results in companies needing to hire more workers to meet the demand resulting in a overall increase in employment.

It also creates a countervailing force that increases employment: people on minimum wage spend their money which creates demand, which creates minimum wage jobs.

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