What are the significant privacy and security risks associated with using a public Bitcoin node for processing transactions, and how might these risks impact user anonymity and transaction security?
Connecting to the third-party node will not compromise the security of the transaction in any way, so you no need to worry about the funds you are sending to someone who will be at any risk if the transaction has been confirmed already.
But the privacy part which has been explained already, your IP will be exposed which can be avoided by connecting via TOR and all the addresses will be known to the node that you connect.
What else it will cause is, you always have to rely on someone to validate your transactions that is why it is advised to run your own node.