Diagnostic fee bait and switch?

So my wife called an appliance repair to have someone come out and take a look at our dishwasher that (thanks to a recent lightning storm) had a burned out circulation pump.

They never told her about any diagnostic fee nor did she sign anything to indicate that she understood or agreed to pay a diagnostic fee. After the tech, who was fine, did his assessment, yup you need a new circulation pump, the office will send you the quote on how much it will cost to order the part and what the estimate for the job will be.

Later that night the invoice/estimate has the wrong part on it, and a much more expensive part, but really the problem is the wrong part. We say um that’s the wrong part, please update this. but we’re like um for the price they quoted we should just replace the whole unit.

So we decide we don’t want the work… now they want the diagnostic fee that they never mentioned but buried on their website it’s “waived if they do the work”

I understand that service calls cost time and money, but if you don’t mention it upfront and don’t get any agreement that we are going to pay it, I don’t owe you anything?

Am I the asshole? What happens if I don’t pay?