That's a very false generalization. Consider this. After a job that was supposed to cost $500, you're presented with a bill for $1500. Or the plumber's helper falls off a ladder and winds up in the hospital. Would you rather have a signed contract in your hand that says the job is going to cost $500 and that the contractor carries insurance? Or would you rather have nothing? With the contract, you send him a check for $500 and if he doesn't like it, he can take you to small claims. With a written contract, you're in an excellent position to defent yourself, if necessary.
Also, having it in writting protects both parties. It's not unusual for two parties to have different recollections of exactly what the scope of the work was, what was included, what was not, etc. Three months later, when the bill comes, who's to say what each party said, who understood what, etc. You want it in writing not because you don't trust the other party, but to eliminate routine misunderstandings.
I'd refuse to pay. And I wouldn't have these clowns back in the house again, regardless of whether you pay or not. And I'd say you made a mistake by not discussing what they would charge, if anything, BEFORE you let them do 6 hours of work. After they inspected it and determined it was a bad solder joint, I would have said "Since it's a bad solder joint, will you repair it for no charge?" And if they said no, then I would have done it myself. Or if it was something I dind't think I could do and wanted to use them for this job, I would definitely have gotten a fixed price up front, not let them go to work not knowing how many hours it would take or if I was going to pay.
Butch up dude!