Think of the business side of the problem as a collective action problem. You want to pay your workers a living wage, but to do so, you have to raise your prices. If you raise your prices, your business will go to your competitor, so you can't exist if you increase your wages (and your competitor doesn't). Meanwhile your competitor wants to increase wages but doesn't for the same reason.
If the government mandated that when you raise your prices, everyone else had to too, then suddenly increasing wages won't put you out of business! Same story if the government mandated that you raise your wages (in this particular hypothetical).
You're right that both sides of the equation need to be examined, but it's actually a bunch of equations with a bunch of sides.
> If you raise your prices ... and your competitor doesn't
this means your business is less efficient than your competitor's, or you are asking for a higher margin than your competitor. Both means that the business deservedly fail under the raised minimum wage.
if everybody has to raise their prices to fund the higher minimum wage, then it means the original minimun wage was already at the optimal level for your business!
If the government mandated that when you raise your prices, everyone else had to too, then suddenly increasing wages won't put you out of business! Same story if the government mandated that you raise your wages (in this particular hypothetical).
You're right that both sides of the equation need to be examined, but it's actually a bunch of equations with a bunch of sides.