It's not the single browser codebase that hurts the web, any browser with noticeable (maybe 3-5%) market share causes these problems. The main issue is that today, the web as a platform is garbage. It's full of cruft, mostly useless features that cause browsers to be bloated, unmaintainable codebases nobody except 9-figure-budget entities can hope to reimplement. It's an unstable, far too complex platform that wastes billions in unnecessary maintenance work for web developers.
Much of this could have been avoided by knocking some sense into the entities that constantly propose new features and those who implement them as native browser elements instead of JS libraries where possible. Then there's the insanity of preserving unsafe behaviour as default and adding response headers and extra HTML tag attributes to avoid it, so broken sites could stay broken and everyone else had to do maintenance work. I'm not saying Google is the driving force behind this, but it reeks of the same stuff that Microsoft was doing to prevent Word, Excel file formats from becoming readable and writable for the competition.
It's too late to freeze the web platform, it needs to be replaced with something sane and stable, controlled by people with no ulterior motives.