Simpson-semistrict categories $n$-categories could be what you're after: categories $n$-categories where everything except the unit laws holds strictly, generalising one of the crucial properties of Moore path spaces? It's not a specific definition of $n$-category, but a strictness property which can be applied within various definitions.
Carlos Simpson has conjectured that these are enough to model homotopy types; Moore path space show this in dimension one1. I know very little about the details of this myself, I'm afraid, but what I have read about it is mostly from these sources plus their links and discussions:
- Simpson, Homotopy types of strict 3-groupoids.
- nlab: semi-strict $\infty$-category
- nlab: Simpson’s conjecture (I can't figure out how to link this directly; the single-quote in the url seems to confuse markdown)
- n-Category Café: Urs Schreiber, Semistrict Infinity-Categories and ω-Semi-Categories
I believe several people have been making some progress on it recently; eg Makkai mentioned some results along these lines at the latest Octoberfest.