Which minimum wage jobs do you know of that aren't in the service sector?
Do you think Amazon, Starbucks, Burger King and Pret-a-Manger will be forced into financial difficulties through paying their staff 3 quid an hour more?
Paying their employees better wages reduces the need for tax credits and helps reduce government borrowing.
It's not £3 though is it and you're not looking at this with a business head.
£3 for the lowest paid workers (plus employers NIC, increased pension contributions etc etc) then lowers the gap to the next tier, which has to be maintained or it's not financially disparate to represent the additional responsibility. Basically these costs steamroll - it's ends up not viable. I understand your point that these companies can afford it, but what shareholder/owner would back that decision with their own money when they will get little direct return as a result, plus make themselves less competitive than the other companies.
As for your point about reducing government borrowing -that's a bit disingenuous too. Higher wages will reduce corporation tax - so it's not really as simple as saying it will be a financial benefit.