[1] http://www.bloombergview.com/articles/2015-01-23/high-freque...
This is available currently in the form of dark pools. They don't tend to work out as liquidity is much lower on them and bid/ask spreads are wider.
That may not be true if ALL venues were required to be private, but that would be a pretty dramatic change (and exactly the opposite of what we have now, which is a requirement to publish and meet other published prices).
If you're a big institutional investor though, and you know that buying a bunch of shares will move the market for sure, well, it might be worth suffering a wider spread. As long as that spread looks small relative to how much you think you'll move the market, it's not a bad deal. And if you already have a subscription to the pool you might as well try it before you get the algorithms involved to buy on the open market.
Dark Pools were meant to provide that, but it turns out that in practice it didn't work (because no one wants to provide liquidity in that environment) and Dark Pools ended up resorting to either letting liquidity providers in, going out of business, or extremely scammy things to keep up the ruse.
So if you are using a liquidity providers it is evidence that you do not want to wait. If you did want to wait, you yourself can just put the order out and provide the liquidity to others.