One of the problems with inviting institutional money into the crypto space is that institutions (like banks and hedge funds) are largely the cause of many of the market's troubles. If things start to fall apart, they will want to hedge against losses, which might include pulling out of crypto and parking money into "safe" investments. So, yes. The more entrenched financial markets get into crypto, the more correlated we will become.
However, I think this will only occur up to a point. The reason we are so much affected by the market is that Wall Street is largely in fiat and making their way into crypto. In the long-term, once they are largely into crypto and partially into fiat, we may not see the same problems. The built-in features of crypto would tie the hands of institutions. For example, we can't print more BTC to bail out failing institutions.
In other words, it's only during this transition period that we may correlate with the market. Once institutions are largely invested in crypto, we may be more resilient, which is actually a benefit to the institutions as well.
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If only retail investors were in crypto, they'd also be likely to pull out during trouble. But because of the rules they're operating under, institutional money has to ditch the riskiest assets first.
As for the rest of your comment, I hope you're right.