These three aspects of monetary media not only are distinct, but actually have conflicting priorities - which is why they are almost never the same medium. If you try to create a crypto that does these three things well it will be a crap crypto because you'd have to create nasty couplings between conflicting objectives:
Payment systems need to be currency agnostic whereas blockchains have to be currency native
good stores of value need to impose scarcity to deflate prices over time whereas good currencies need to inflate liquidity to keep prices stable
currencies must be definable against a heterogeneous background of both payment systems and collateralising assets, whereas payment systems only facilitate the clearing of a trade and represent neither a currency nor an asset
This is a very good analysis. That said, block chains can take on the role of "small community currencies" where they also perform the function of a payment system. Using them as a store of value/speculative asset (which is most of crypto these days) is a totally different role. But you can have many block chains that perform many different functions - what is still not present sufficiently, because of bitcoin's domination, are distributed exchanges that can connect different chains, and as such, form a larger payment system.