That's correct. No Omni hotels are Host properties.
And you're right on the other point too, or so it seems to me. The contracts with 'Host' would probably be individual contracts with the companies on Host properties, technically speaking.
Per the debt/equity financing conundrum, my limited financial imagination cannot foresee how they can practically effect that transition, i.e. from latter to the former, unless what we thought we heard about the hotels footing the bills for installs. Seems like without that, they will always need to go prostitute themselves just to get the money to fulfill whatever contract comes in.