To some degree there will be cases like that, but that is not really what happens on a macro level though. In aggregate, there tends to be no statistical correlation between the minimum wage level and employment levels. Companies try to be efficient, so they try not to have any more or less people than they need (based mostly on demand). The desire to pay as few people to do as much work as possible always exists. Higher minimum wages may put some extra emphasis on this, but it's likely that the extra demand from consumers having more income balances this out.
What's your source for this? If there's no statistical correlation between the minimum wage and the unemployment rate (and I thought I had read otherwise) then I suspect the statistics aren't telling the whole story.
Your explanation of what's going on at the micro level contradicts my experience as a business owner who's been signing paychecks for many years. All businesses have inefficiencies, and in general the larger the check, the more scrutiny the expense will receive. Internally this manifests as having higher expectations for a higher paid employee, and prioritizing automation, offshoring, or other business decisions when labor costs get too high. Externally, when we expand our relationship with a client, we fully expect that it will receive more scrutiny from various stakeholders at the company simply because the check getting written to us is bigger, even if we deliver proportionately more value.
