At least that's been my experience. In fact, I've even encountered problems while using Chromium and Firefox on Linux, just because some sites didn't like the user agent.
In short, to use the modern web you need a modern browser, and modern browsers are very leaky and fingerprintable by design.
> Maybe VPNs should start to offer “browser anonymization” as a service.
The problem is that they'd need to render the website server-side and then serve it to you. That has their own problems, as the VPN provider now has total control of all web content you see.
That already exists, by the way: https://www.puffin.com/secure-browser
I'd say the most realistic options to avoid browser fingerprinting is either using Apple hardware or sandboxing the browser inside a virtual machine. And it's better to use Chrome because it has the most users by a large margin. Firefox, Brave and the new Mullvad browser do implement some anti-fingerprint mitigations, but they have few users so you'll stick out more.