Is it important to diversify by holding investments with multiple investment management firms?

Let’s say a person wants to invest in a simple total-market stock index fund, and nothing else (besides bonds, etc). Should that person hold total-market stock index funds with multiple investment management firms rather than have their entire investment with one firm?

For instance, by holding total-market stock index funds with Fidelity, Vanguard, and Schwab, as opposed to just one of them.

I’m not talking about diversifying the allocation of the portfolio itself, since all of the index funds would be almost identical, and even one broad index fund is already diversified all by itself. Rather, I’m asking about reducing the risk of holding all one’s investments with one firm, if there is any risk to doing that.

No, that serves no purpose. The security you invest is only held by the broker in kind or in you name. If they go bankrupt, your security is safe because the broker had no stake in the investment but facilitated the sell. If you are happy with your investment strategy, just buy assets from one firm, preferably with the cheapest trades! BofA Direct Investor has $7 trades if you meet certain requirements.