I've seen meter fitters test polarity at sockets before and after replacing the meter, to ensure the wiring is the same after the meter is replaced. Any wrongly wired socket would then show up before hand. It's good they check, we did a site of houses about 9 years ago, at the start we would do a second fix, then dead tests, then connect the tails to the meter and complete the live tests. About half way through EDF made some changes to how things were going to work and that they wanted to connect the tails at the same time as the meter was fitted, after being given a copy of the results of the dead tests. Trouble was a total of 3 houses ended up being connected backwards by the meter fitters. We'd then have to call them back to correct it.
The appliance man was probably just trying to ensure safety. Without a good earth a fault on an appliance could become a hazard. Many white goods are also fitted with substantial mains filters, the earth leakage from these is enough to cause an uncomfortable shock from the metalwork if it is not connected to a properly earthed supply. If he was to carry out the repair, and then inform the client of the issue, the majority of people wouldn't care, after all, the appliance is working again. The fault then gets left to cause trouble in the future.
The appliance man was probably just trying to ensure safety. Without a good earth a fault on an appliance could become a hazard. Many white goods are also fitted with substantial mains filters, the earth leakage from these is enough to cause an uncomfortable shock from the metalwork if it is not connected to a properly earthed supply. If he was to carry out the repair, and then inform the client of the issue, the majority of people wouldn't care, after all, the appliance is working again. The fault then gets left to cause trouble in the future.