Chromium isn't just gratis software ("free" has a different meaning), it's open source software. There is no implicit expectation that downstream users can't change it any way want[1] and redistribute the result, that's the whole point of the open source license that Chromium is released under.
Chromium is distributed under the 3-clause BSD license, so I totally agree with you that distros can do whatever they want with it (more details here: https://www.chromium.org/chromium-os/licensing/ ). I'd imagine many people that work on Chromium would agree with this and are happy for distros to do what they'd like. If Google wanted to be pushy with the software, it could do some other kind of licensing saying people couldn't modify it and still use the Chromium branding, but they obviously chose not to do this.
My take from a business perspective is that Google produces Chrome and Chromium for a number of reasons. Good will to the community (with how permissive they are with the license), and having a stable platform to be able to build things like GMail and Search on-top of. But there is also the Ads side that benefits from Google Search being the default.
So I guess there are really many benefits for Chrome's existence, and Google Searching being a default is only part of that. But I still stand by my original post and reasoning.
1. Within existing legal boundaries, of course