Here's where theory and reality conflict.
I wouldn't say that it's theory and reality conflicting so much as it is ideology and reality, or possibly desires and reality. The fact that, the higher price a good or service is, the less that's demanded is so well established that it would hardly be worth arguing about if it weren't for entrenched interests who benefit from claiming the opposite. The only exception I know of to this general rule are "other than normal" goods.
So, is labor an other than normal good? I don't think any economist would claim such. An other than normal good is something like the potato during the Irish potato famine. What happened is that the demand of potatoes went up as the price went up. That seemingly paradoxical result was because all the alternatives were even more expensive. People reverted to potatoes because it was the only way they could get the needed calories to stay alive.
So for labor to be an other than normal good there would have to be no alternatives. Unfortunately for proponents of minimum wage, there are. Hiring highly productive people to replace more low-skilled people is one alternative. Outsourcing to other countries without a minimum wage is another. Increasing automation is a third.
Let's try a thought experiment. Suppose we're trying to decide between two minimum wage laws, one setting it at 10 credits per parsec (Yes, I know.) and the other at 11 cpp. If increasing the minimum wage truly didn't reduce employment then the law setting the minimum wage at 11 cpp is clearly better because low-paid people now have more money to spend, and employers continue to make a profit. Employers continue to make a profit because otherwise they would go out of business, which would reduce employment, and our assumption was that wouldn't happen. Now consider a law setting the minimum wage at 12 cpp. It's clearly better than one setting it at 11 cpp. A law setting it at 13 cpp is clearly better than one setting it at 12 cpp. This can continue, considering higher and higher minimum wages, until each worker is granted an infinite amount of money per parsec. This is clearly impossible, therefore the minimum wage law must affect employment.
There's no concrete evidence that the minimum wage reduces employment.
That's not true. There's plenty of evidence that it does--most notably the 1983 Brown study.
Some studies go one way, others another,
Some studies are seriously flawed. For example the 1992 Card and Kreuger study is flawed for studying same-store employment. Since most new businesses fail before two years pass, studying same-store employment discards lots of important data. There's a humorous example why this is true in the proof that no one is ever executed on death row. Imagine an economist trying to determine how many people are executed by taking a survey of people on death row one year. Then the following year they go back and survey those same people. If the people are no longer extant, then they discard that information. Thus they arrive at the conclusion that no one was executed because all of the people in the final data set were not executed during the year.
Other studies are flawed because they look at one measure at the exclusion of others. For example, unemployment is defined as the number of people looking for work divided by the sum of people looking for work plus those having a job. Clearly the desire is for unemployment to go down. However, a myopic focus on the unemployment figure can conceal negative consequences. The 1983 Brown study I mentioned earlier found that teenagers leave the work force in response to increases in the minimum wage. If those teenagers leave the work force then the number of people looking for work may fall, despite there being fewer people having a job. The unemployment figure could fall, but no one concerned with the welfare of teenagers would claim that their state has improved.