Bitcoin was solely built for the purpose to eliminate the third party
Bitcoin eliminated needing a third party to transfer, store or generally use your own money. And it does that perfectly.
and yet we let the third party[i.e. exchanges] decide the price of Bitcoin today.
First of all exchanges have nothing to do with bitcoin and elimination of third parties as I explained above.
Secondly exchanges don't decide the price, the people who trade on those exchanges do.
Bitcoin & Cryptos come with a downside of manipulation as for example:
Altcoins are all manipulation but bitcoin manipulation is small and is shrinking as the market grows.
EXCHANGEs which in turn creates a fake situation in the market about the demand of Bitcoin which then taken into account by charts,
It is nearly impossible for the exchange to create anything fake that affects the market. Lets say the exchange places fake buy orders, it is possible that people sell into those buy orders that means the exchange has to pay them fiat so it costs them money and you can no longer call the order "fake" if they did pay money!
This is what happens when you have big Bitcoins in your hands.
Another thing that happens is that the whale sells the bitcoins he has but the price doesn't go down and he loses money. For example last week a lot of whales did that trying to prevent the price from rising above $40k and lost millions of dollars.