The ZX calculus was not designed to be a programming language, or a language in which to specify things at a high level. It is a language for reasoning about things on a relatively low level, albeit without digging into the actual physics.
What it was designed for is hinted at in your observations, and by the name of the ZX calculus itself.
It can describe individual quantum gates: so it is a notation for circuits (and other quantum procedures — one of the things the ZX calculus was explicitly designed for was to analyse measurement based quantum computing [arXiv:0906.4725]; and Dom Horsman and I showed that it has a very close connection to surface code lattice surgery [arXiv:1704.08670]).
It is a calculus : it is a notation in which you can actually do calculations. If you're good with reasoning about commutation relations, you can do this to a limited extent with ordinary circuit diagrams; for the ZX calculus you can in principle do this entirely with diagrams.
There are people who are working on higher-level ZX descriptions of procedures on multiple qubits — eg. [arXiv:1905.00041] — which might become suitable to actually program with if developed further. But the existing version of the ZX calculus is more suitable for an intermediate representation of a compiler, or indeed performing computations or analysis by hand, than as a programming language.