By that logic, wouldn't it be a Pagan holiday regardless of how it was corrupted by Christians? I mean, if how the holiday is portrayed today cannot change its Christian core, certainly it would stand to reason that how the holiday is/was portrayed by Christians wouldn't change its Pagan core.
Christians 'graduated' from Judaism when Jesus died. It's not as if it was 'stolen'. The pagan holiday was eventually extinguished by Jews and replaced with Jesus by Christianity. This occurred around 300 AD. Santa Claus and trees are just fun counterparts they decided to keep and is not deemed a requirement by Christians to celebrate.
Saying that it's pagan origin defines it as a pagan holiday today would be to say that Christians should be considered Jews because of their origin.