There are many roads to the cheese. Cold calling, for example, works. I've never done it and if I suggested it to my (very successful) clients they'd laugh in my face, but it is an enormous business. A lot of businesses can get away with terrible websites for the same reason I can get away with terrible business cards: 99.8% of the business is elsewhere. If you're Nobu your website could be done by a 4th grader in Fl…
what designers like about designs is in many cases orthogonal to their effectiveness in convincing customers to enter business relationships You have there, in a nutshell, the difference between the outlook of good and bad designers. Bad designers care about whether they like the design. Good designers care about whether the end-user likes the design.
Should a site look good? Absolutely. It never hurts anything to look better. Can a site look ugly? Absolutely. If your business is strong enough, there's no amount of bad design that can hurt it.
The main thing though, is that more attractive designs don't necessarily convert better. People think they do, because you're turned off by ugly designs, and for sure, if you've got two sites up that share the same interface, and one is prettier than the other, the prettier one will likely win your business. But if you take those same sites, and have better copy on the ugly site, it may well convert better. If your pretty design is hard to use, it may well lose.
So yeah, if your web designer cares more about the end user or themselves than whether the design is effective, then you're paying for the wrong thing.