Chromium has no auto-updater on Windows and macOS. Unless you have a package manager that compiles updated versions for you, you're better off using Chrome.

I use Firefox. But, I'd be willing to install a version of Chrome if it came without all of the Google garbage pre-installed. I was hoping that maybe Chromium could fill this role.

There's a project that does exactly this. (I can't comment on the quality or the trustworthiness.)

https://github.com/Eloston/ungoogled-chromium

I have used it in the past but now I use Chromium instead as a Chrome replacement since it is more stable than ungoogled-chromium.

I use Firefox as my primary browser and Chromium for testing things for Chrome. The only downside to Chromium is you have to update it manually by downloading the package and dragging it to your applications if you use Mac. Other than that, it works perfectly. I trust Chromium more than Chrome if I want to let it run in the background.