I think all those things are quite unnecessary. Assets can't be a safe haven all the time. There are times that cash can be a better option as a safe haven, and sometimes gold is a good option to go with. There are times you will check and a particular asset will be doing fine while the majority in the market are falling. You can decide to buy that particular asset so you will be able to save your money and not lose it.
So in this case of cryptocurrency/bitcoin, I don't think it's necessary creating what you have referred to as S&P 500. Cryptocurrency is different, and it doesn't correlate with stocks just as you have said, sometimes when it does it's just a coincidence. They don't correlate in any way, neither does it correlate with gold. The only thing bitcoin correlates with its same cryptocurrencies such as Ethereum.
I think it's very necessary you understand the point from which I'm heading from, cryptocurrency fundamentals is quite different from stocks. But it clearly looks like the market sentiment is beginning to mirror each other.
There can be times when they mirror each other but in reality they are completely different from each other and bitcoin don't follow The stock market. In the past also there used to be a reverse trend between these two, when the stock market used to fall bitcoin rose in price and when former rose the latter fell, this is just a speculations all the time, the market sentiments changes and a lot of factors make the Price rise or fall we can only try and find these relations and still bitcoins will be unpredictable.