Train: yes, for now.
Host: depends on the scale. At a small scale a wealthy individual could easily build a rig in their basement to host one of these things. At larger scale any cloud company could do it, and many already have the compute on site. At large scale this is true... again, for now.
What you say only holds (in the absence of a state oligopoly) if two conditions are met: (1) AI performance does not asymptote any time soon due to running out of training data or other scaling limitations, and (2) these companies are able to stay at the frontier.
There's little to no moat, so staying at the frontier will be a game of investing massively in compute, talent, and R&D, and they can never stop.