How is the interaction between an Hardware Wallet and the blockchain?

I tried to understand the principle how the hardware wallet sends transactions to another Bitcoin address since it is not connected via full node to the blockchain.
I found this answer: How do hardware wallets communicate with the outside world?

Do I understand it right that every hardware wallet company like Ledger, Trezor or Shiftcrypto run a full node (e.g. bitcoind) in their company network and every hardware wallet, which is connected via the locally installed company’s software at the sender’s computer just sends a request to this full node like this?

bitcoin-cli createrawtransaction "({"txid" : "mytxid","vout":0})" "{"myaddress":0.01}"

If so, do companies run it’s own (closed source) full node which works in a different way? The only exception seems to be BitBoxBase from ShiftCrypto, which is open source.

But basically, the principle of the transaction handling between the hardware wallet and the blockchain is always the same, right? So every company run it’s own full node. Only the transmission “protocol” between the hardware wallet and this full node is different and non-standardized, right?