Can Google help me root my Android device or do I have to through the mfg?

That's a damned good question.

I'll say that I've had an (apparently non-rootable) Android device, purchased under the explicit understanding that it would be rootable.

My interest and enthusiasm for the Android "ecosystem" has ... cooled markedly.

I'm looking forward to what Purism, Pine, and/or ReMarkable come up with in future.

As a practical response:

1. You're generally on your own.

2. If you cannot find a root procedure on mainstream sites (e.g., LineageOS or similar), almost certainly none is available.

3. Return the device for a refund and check the compatibility lists first next time. Or buy an advertised-open-boot device.

generally this is a no (the exception kind of being pixel/nexus devices) running non official roms and mods requires an unlocked bootloader which is something only the manufacturer can provide, some devices like for example huaweis mate 20 pro have no option for unlock/root at all (unless you already got your bootloader unlock code before huawei closed the service)

that doesnt mean yours is impossible though, let me know the model and i may be able to help