Chrome Enterprise (which, contrary to what the name might suggest, is not a paid enterprise offering) offers management tooling for managing flags across many devices. Here's the flag for DoH: https://cloud.google.com/docs/chrome-enterprise/policies/?po...
but Chrome Enterprise means it has to connect and phone home to Google all the time anyway, no? That defeats any potential benefit of having more control over the browser.
Given this is a "free" offering, the data being mined finances this service.
If parent wants to manage Chrome configuration across N devices, Chrome Enterprise is a good tool for the job. They may or may not care if their data is on Google servers or not. They might consider these two items to be two entirely distinct and different benefits.
If parent wants to avoid having any of their data cross Google machines, you are completely correct that Chrome is the wrong tool for the job.
Given this is a "free" offering, the data being mined finances this service.