The amount of institutional investors in bitcoin is still extremely small compared to, for example, any of the dow stocks. While the idea of giving away bitcoin to the hands of wall street billionaires may not sound appealing, it is the best way to stabilize the price. When something like 50% of bitcoins are owned by institutional investors, the price will definitely be significantly more stable.
Retail investors will always buy into manias and push assets up and down like crazy. They have weaker hands. Especially bitcoin investors. As long as people are out there trying to use bitcoin to get rich quick, it will remain too volatile to stabilize and grow as a currency.
The question is: do we really need institutional money to flow into bitcoin?
Of course, it would be great, but Bitcoin if just fine as being alternative asset - used mostly by people, not businesses.
Bitcoin is like stock - available for anyone, without need of contacting any middlemen to purchase it.
Price of bitcoin will be more stable over time, more people using BTC = less volatility.
The answer is yes in my opinion. I don't see how else the price will stabilize. The nature of bitcoin investors is not going to change. Bitcoin is in a self-feeding cycle where volatility attracts get-rich-quick investors, and get-rich-quick investors cause volatility. This all makes it less useful as a currency, and therefore it harms bitcoin's intrinsic value.
You say the price of bitcoin will be more stable over time, but why do you say this? It has been volatile over the past 8 years since it was created, and it has not become any less volatile over time. Why would it stabilize now?