It could be that the administration doesn't trust that they're getting good data from the CDC. Thus they're requiring the data be sent to another agency within the DHHS. I'd be surprised if CDC doesn't receive the info as well but it would have to be forwarded to them. If the CDC is filtering their "output" through a political lens, that's a problem as that's not supposed to be what they do.
You have the power balances absolutely backwards. CDC's director serves at the pleasure of the president and can be replaced at any moment without senate confirmation.
DHHS, the VP, and the President can have access to the raw NHSN data collected by CDC if they think there is anything biased, hinky, or whatever in their interpretation thereof.
Instead a new collection infrastructure and interface was created and all the hospitals had little notice to prepare to shift their reporting. Why?
It is so bizarre that nefarious assumptions come easily. The only reason to move the data collection on infection out of CDC is to filter what CDC has access to, or to simply be capricious in response to not giving the president''s preferred political spin by suppressing data (and seemingly to make life harder for all the hospitals that already have gargantuan data reporting burdens to many levels of government by making us all switch, again).
Last, your implication that COVID-19 death stats are skewed by counting dead people with the virus is pretty silly. Many or most states track COVID vs COVID-related deaths, but it is a distinction with little meaning. It is like saying that CDC skews smoking related death statistics because they count lung cancer in smokers as smoking related deaths when it was really the cancer or complications of cancer that killed them. What upset the apple cart?