Bitcoins value does indeed affect the price of all altcoins since BTC/AltCoin trading pairs are the most commonly used and is usually the standard go to exchange pair on all cryptocurrency exchanges. So can Altcoin break all the relationship from bitcoin and run as a independent currency.
i think it's possible to to seperate them but that's not happening now most exchanges don't trade Fiat pairs and have only a few coins against USDT/TUSD if more and more exchanges start trading vs USD/EUR i think there values could seperate but the regulation are just ambigous now that not alot of exchanges can get into it.