It's indisputable that the two are happening -- the labor pool has shrunk, and real wages have shrunk too. And it is consistent with economic logic that greater business expenses will lead to inflation. But economic models also predict that scarce labor should lead labor to command higher wages. How those two effects net out is an empirical question -- no model can yet answer it definitively. (The models and the data are, however, both clear that freer immigration leads to greater total world output.)

Also inflation is driven by labor squeezes outside of the United States. (Freer immigration into the US would actually worsen that problem, but negligibly, since US immigration is on the order of 100K people annually out of a world total around 8B.)