If you keep altcoins in hardware wallet, how dependent you are from the model of hardware wallet?
To stay on-topic, I'll answer this for Forkcoins (but it applies to altcoins too). You can use your seed words to restore all private keys. The easiest way to do it is using
Ian Coleman's Mnemonic Code Converter.
DO NOT use this online! You should use the offline, air-gapped, on a system without storage space that runs from a Live Linux DVD with the curtains closed. Even then, you should probably consider your seed words compromised, just in case.
Btw, I also wonder how it works technically, when the same seed "remembers" and "keeps" many altcoins while every altcoin has its own blockchain? Every altcoin then should have its own seed. But somehow all of them are stored in the wallet with only 1 seed. How? :-)
The key word is Derivation Paths, which convert your seed words into private keys. You don't need a blockchain to do this.