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 believe that when Bitcoin prices have risen to a steady state, we can see that Altcoin is showing an independent market, which requires global recognition of BTC. We can't see it now, I predict that we can see this happen within 10 years.