The default fonts on Windows, Mac, iOS, and Android are the result of millions of dollars of R&D.For quite some time, Apple was using Helvetica, a typeface that has objectively awful (edit: legibility) properties.
Meanwhile, some of the web fonts you see have had a very large amount of money spent on making them and world class font designers working on them.
So I don't think this argument carries much weight.
Custom fonts are far more likely to have issues with kerning, hinting, or subpixel rendering.
That depends entirely on the quality of the custom font.
As another example, for a long time on Windows you could get strange results with font stacks that used Helvetica/Arial/equivalents, because someone's random printer font would end up taking precedence over the system font you intended and result in your page looking atrocious. If you didn't know that one and used some widely recommended font stacks based on default system fonts, you would get awful results for a significant number of users.
That's what Github, Wordpress, Facebook, and Twitter do. They all have distinct brands without custom fonts.
They all have strong brands without any reference to visual design at all, other than arguably their highly recognisable logos.
However, the other 99.9999% of businesses you'll come across don't. If you're not already a world-famous unicorn brand, a well-designed and consistent look and feel does affect brand recognition and, in some cases, have very practical effects on usability.
