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.
If it can, the market will change completely. But I think that is difficult to happen. As you see, altcoins are too dependent on ETH (erc20, smart contract, ...). If it can break the relationship with the BTC, it still depends on ETH. Nowadays, many coin have technology independence (XRP, NEO,..), they are gradually becoming independent.