The problem with using public index funds to mimic these managers asset allocations is the higher volatility of publicly traded assets. For example in 2008 widespread panic caused people to sell their mutual funds and flee to cash which widely hold a variety of assets, so all of them suffered. This included publicly traded Real Estate Investment Trusts (REITs). At one point they were selling for 80% of their on paper asset value, which was entirely irrational. Many money managers that typically invested privately in commercial real estate because of that volatility took that opportunity to buy public REITs at a discount.
That being said, you can definitely do a decent job of adding asset class diversification with modern ETFs (and the market for these keeps growing). There are now infrastructure ETFs, REITs (Vanguard's REIT is really well priced), covered call ETFs, etc. Just don't expect to get exactly the same results of these managers and expect to have to handle larger volatility.
That being said, if you can weather these market movements, and you are saving regularly, volatility can be a good thing, if you can take advantage of lower asset prices with ongoing contribution then that is great.
I know these are naive questions, but here goes: do those pension funds differ greatly from large university endowments? How do those differ from what one can achieve using a combination of cheap index funds? Ramit Sethi famously recommended an allocation based on David Swensen's experience managing the Yale endowment. You can find index funds to match that reasonably well. How different will that be from what Yale or a pension fund actually does?