There are two types of apps: 1. the ones that professionals use and 2. the ones that consumers use.
for 1. they don't care if it looks native, as long as it works and is performant e.g. DAWs, Video Editing tools, Trading, etc.
2. likewise I don't think it matters that much.
my guess is the myth came from OS makers.
Moreover, web UIs tend to be less sophisticated and less power user friendly, due to HTML/CSS and browser API limitations. This unfortunately often carries over even to non-browser-based applications that however use a web-like UI.