You claim raising the minimum wage would never raise prices nor cause unemployment to rise. Ok, let's set it to $1000 a hr. Do you really think that would have no effect on either of those? Maybe there's some range on increase where the effects are offset by other effects but it's not lying to believe those effects will become real at some point.
Realistically speaking, though, most arguments are over minimum wage hikes that only affect profits and won't even have a minimal effect on prices. $10 -> $12 isn't even going to cause prices to rise, let alone have an effect on employment. It'll go straight from profits to paychecks.
I can think of only one instance where raising the minimum wage actually affected employment, actually, and that was because it was set at first world levels in a third world environment.
I think you're wrong in thinking that businesses will just give up, and take lower profits. No. They're going to cut hours, fire anyone they can, etc. to maintain their existing profit levels.
But no, they don't just sit there and take it. They pile money into think tanks and advertising campaigns in order to try to sway public opinion (see above).
One of my favorite tactics is when they pretend that robots that do minimum wage jobs all cost $minimum wage + $1 and raising wages is simply going to make them all go out and buy those magical robots.