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.
we need more stablecoins for that. I have seen many activities in this field recently, that should significantly help altcoins to go more independent from bitcoin. Time will tell