Is that because once redeemed the private key becomes public and anyone can then use it?
Your private key will always remain private unless you leak it by mistake. It doesn't matter if you used it create a signature (create a transaction and spend your coins).
If your address doesn't show any forkcoin balance it is most probably because you spent your bitcoin before any of those forks took place. For example BCH fork was on height 478558 (2017-08-01) and you would only get BCH if you had spent your bitcoin in block 478559+.