Ok. I can see this works for Bitcoin only. But what if I want to preserve my balance from other Bitcoin forks (Bitcoin Cash, Bitcoin SV, etc)? I have read somewhere that such forks use the same private key. I don't want the balance from other coins to move into the same wallet address. Instead, I'd prefer to split my BTC into a new Native SegWit address, while I send other coins to separate addresses. Is this possible?

Yes. Your forked coins will not move when you move your Bitcoin Balance to a new wallet.
All you have to preserve any forked coins you want to someday is preserve the private key where the coins were held at the time of the fork.
As you have correctly noted, it is best to move all your coins to a new wallet address before attemtping to claim any of the forked coins.
To claim those coins, like Bitcoin Cash, you'd have to download a Bitcoin Cash Wallet, and seed it with the preserved private key.