I was not aware of this issue, and when I had looked into things earlier on my own, I had discovered a reddit post from what I thought was the Ledger CTO that said

The only feature that cannot work without our servers being offline is the firmware update functionality

I also knew that while I could never actually get to the private key (or at least practically) stored/generated on the wallet, with the BIP39 Mnemonic and optional passphrase, I could regenerate the private keys.

So with the above information, I figured I was good, but didn't take into account a panic sell situation. One of the reasons I keep crypto on a HW wallet is for the added psychological benefit of preventing panic sell situations.

This still, however, conflicts with how I thought this worked - I still didn't think that Ledger/Trezor's servers were needed if I was simply storing a key on my hardware wallet. If I use any wallet - say, Electrum, MyCrypto, MyEtherWallet or anything that works with a Hardware Wallet, are you saying that still depends on the parent company's servers?