Keeping wallet addresses outside bitcoin core
Does using this architecture have any disadvantages over running a full bitcoin core to keep addresses?


Yes.


  The "bitcoin core" will be some trusted public node that I don't control.


That is a huge disadvantage.

As Satoshi Nakamoto wrote:


  What is needed is an electronic payment system based on cryptographic proof instead of trust,
  allowing any two willing parties to transact directly with each other without the need for a trusted
  third party.


(my emphasis)

Every time someone tries to undo this they are kicking the founder of Bitcoin in the teeth!



Other references


Thin Client Security