On the face of it, I could see how people would agree, but there is more to it IMO.
I run a firm of accountants. We have someone that started as a trainee, but failed his apprenticeship. He's a good lad and does his job well - he's just not interested in any further training at this moment in time.
His job does have some stressful periods but generally he's in the door at 9am and out again at 5.30pm.
Up until recently he'd have been miles away from minimum wage, but that minimum wage has now been brought up to his level so he's now officially on minimum wage.
I considered whether people would likely choose a supermarket job over a professional role on minimum wage but ultimately my colleague works 37.5 hours a week, Monday to Friday, so no anti-social hours, etc.
He then still has the opportunity to further his career if he decides he wants to later.
Also, I don't think that minimum wage has increased to a rate that is too high, as even being on minimum wage still won't allow most people to live in their own home in most parts of the country. I just see it that minimum wage has just been way too low.
At the old minimum wage people still couldn't afford to rent or buy their own places, so would likely have to rely on Universal Credit to top it up or pay a housing element.
With NMW going up, this should (in theory!) reduce the amount of UC top up that is required.
Unfortunately there will be other knock on effects like other costs going up. The supermarkets could, conceivably, swallow the increase in cost through their profits (but won't, because they have shareholders to answer to), but your family run independent restaurant probably won't, so their prices will have to go up.
Those saying that they'll be earning the same as someone cleaning toilets - would you want to do the job for £12.21 an hour? I know I wouldn't! I'd much rather have an office job at £12.21 an hour, than cleaning a shitty toilet. Quite frankly those people deserve more!