You should've broken up the soil before planting the grass.
Personally, I'd get a pair of aerating sandles, as they are likely to last longer than that ^ contraption. Probably worth adding some organic matter too, as that'll help improve the quality of the soil and break it down.
Looking into breaking up this hard soil without totally uprooting the new grass, what are my options?
Thinking some kind of tiller/miller/cultivator. Something one step more than poking holes, but not further than scarifying?
Maybe something like this?