They do. A large percentage of the population has 401Ks, and many public sector pension plans have equity components. Housing prices are also assets, and inflation in that sector has priced out many people from affording homes, despite low financing costs.
> Full employment is going great -- the job market is tight
We are nowhere close to full employment. We're 3M jobs lower than pre-COVID in the U.S [1], and according to Keynesians, Monetarists, Monetary Keynesians, or whatever hybrid form of wishy washy economics that has used the Phillips curve as policy guidance, this inflation should not happen.
> In the larger sense, the fed's hands are tied, unless their legal mandate is amended to include "not creating asset bubbles"
The Fed's "dual mandate" presumes that (based upon the Phillips curve) there is a sweet spot between full employment and inflation. The correlation is entirely, completely broken [2].
[1] https://fred.stlouisfed.org/series/PAYEMS [2] https://www.nber.org/digest/sep19/phillips-curve-still-usefu...