They got... different reforms. Augustus's solution was to distribute provincial land (funded by taxes on inheritances and slave sales) to retiring troops after set (long) enlistment periods, which merged the broader issue of land reform and veteran benefits, both of which were key to the political unrest of the Late Republic. The Senate, in the short-term, didn't have to share its power with the dreaded poors - in fact, the Popular Assemblies were almost entirely neutered, which was a key demand of the ultraconservatives in the Senate during the Late Republic.
The thing is, once the Senate (effectively) gave up their power (and any who weren't keen on doing so replaced by lickspittles by Augustus), all of their prior conservative demands meant jack shit. The Senate, instead of becoming the sole instrument of power that it dreamed of, became a rubber-stamp legislature and advisory council to the Emperor, while ambitious men increasingly climbed through the ranks of the (newly permanent and professional) Legions, which significantly altered the political discussions of Rome.
With power more nakedly in the hands of the men with swords, and those same men being settled largely in colonies in the provinces, the political discussion turned from land reform for Italian magnates, to broader issues of provincial inclusion in the Roman polity. Coincidentally, this also happened as the Roman Empire became increasingly urbanized (not just in the city of Rome itself) and wage labor and mercantile endeavors began to develop as a serious alternative to subsistence farming in many areas, along with a drop in the birth rates of Roman citizens, all of which also reduced pressure for land reform.
The aristocracy continued to be a problem, but a much reduced one for the next ~300 years of the Empire, since the levers of power and wealth were funneled to new, rising bureaucratic and mercantile classes, and the traditional land magnates were increasingly marginalized - especially since they no longer actually had the power to obstruct or construct legislation in their favor to enhance their wealth and power. This despite essentially getting all but one (supremacy of the Senate) of their core demands of Late Republic politics fulfilled. Funny how that works. All the policy successes in the world mean nothing without the power base that causes them.
Unfortunately, as the Empire fell apart in the Crisis of the Third Century and became increasingly rural and demonetized, and the central government weakened, land became the core of economic power again, and so the fundamentally unresolved issue of aristocratic land ownership allowed the aristocracy to reclaim massive political power and reduce the common folk of the Empire to a debased state of borderline slavery until the Empire fell entirely.