I need to see a citation for that. My gut tells be that if people had mostly accurate at-home tests they took ~daily, it would actually reduce spread massively, especially when it can be asymptomatic.
I also expect that compliance wouldn't be all that high for at home testing (people skipping tests, deciding they still feel good enough, etc). It doesn't mean we shouldn't try to establish that access for people that would use it well though.
Also, there's a difference between working to increase testing so that more people do get tested before a holiday and making that the center of your message. They need to be careful to not mislead or give the impression that they are misleading, but they could limit the messaging to a statement about working to expand testing.