How does a hardware wallet interact with the bitcoin network?
No.
The hardware wallet company sometimes runs a node, but you could just as easily use a client like Electrum instead, and use Electrum nodes instead with your hardware wallet. All the node is used for is checking for transactions and publishing transactions.
The wallet software will use the xpub from the hardware wallet to check for payments people have sent to your addresses, using one of these remote full nodes.
Now, let's say you received some Bitcoin and you'd like to spend it. Your wallet software will generate an unsigned transaction on your computer, then it will send that unsigned transaction to your hardware wallet for it to sign, then the wallet software will use the full node to publish this signed transaction.
So like the answer you linked to pointed out, all the full nodes are used for is 1) querying for existing blockchain data, and 2) publishing (NOT creating) an already signed raw transaction.