Yes the CMTS does it pretty much like you describe.
Though some have managed to flash modems with a compatible firmware build (see other threads regarding this).
At present ATPIA providers don't have a system or method of doing this on their own. And the incumbents have been a little sly with the CRTC and inserted stipulations that they won't push firmware on behalf of their ATPIA clients on purpose as part of their competive advantage strategies. This leaves end users in a no firmware update even on compatible approved modem models with older firmware in particular the Docsis 3.0 units.
This has forced some to seek out 3rd party modem firmware fixes as mentioned above.
Its quite sad it has to be this way. But thats the nature of the beast.
What happens down the road at the next CRTC tariff rounds in the comming months ahead that might or might not change this is anyone's guess.
And Thank You for a little more of a technical overview of the CMTS breakdown swiftec --
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