Those differences, v (public) stock/bond only, might fall into two categories.
1. Things you can more efficiently invest in, or invest in at all, only if you have a large amount to invest
2. Forms of (attempted) diversification which might or not might have value.
It seems to be direct private equity and property would tend to be in the former category. I do not accept the opinion sometimes given or at least suggested on this forum that public stock is a strictly superior investment to private equity or real estate (and everything else

). It's just that the latter two present some serious obstacles to smaller scale investors. But if 20% real estate is $10's-$100's mils or more then you're not fielding calls on the proverbial clogged toilet in the middle of the night, nor do you have to concentrate in a few properties. Likewise there are potential advantages to private companies over public ones in particular situations, especially if you are not paying high fees over and above the management of the companies themselves (obviously with a stock index fund you are still paying the cost of managing *the companies* part of the expense which results in net income, just not much additional fee to manage *the fund*).
Hedge funds and private equity *funds* would be justified more on point 2. The very rich are still paying fees. So it's still subject to the argument that it's just the insiders in the best such funds who can create real value, not necessarily investors on the outside paying fees, and the most persistently successful funds may tend to kick outsider investors out eventually. OTOH most anti-HF argument here is by people who wouldn't be eligible to invest in them anyway, and often via simplistic arguments like 'they didn't beat the S&P'. It's not actually a solid fact that HF, VC, PE funds add no value, again especially if hiring your own good analysts to evaluate them cost a negligible % of your assets per year.
Which is the basic reason why it might make sense for 'offices' of very wealthy families to invest differently than X/Y public stock/public bond: the cost of the 'office' could be quite small in bps per year, if the fortune is big enough. And no reason to suppose such economy of scale doesn't open *any* worthwhile investment opportunities that wouldn't be open on a portfolio of a few $100k or a few $mil. Although as already mentioned, this is a big reason why the allocation isn't directly relevant if you are running a much smaller portfolio.