Seems it should often be possible to tell if the deadline is real or artificial. Perhaps by asking a question or two, but if it's something like "we're planning to start a major new project soon and we want the new head (i.e. you) to be there at the start", that seems likely to have come up early in the discussion of "what would I be working on".
Also, the larger the company, and the less specialized the role, the higher is the likelihood that "if you're above their competency threshold, then there's some team somewhere that needs more people even if they hire someone else for this role".
Also, the larger the company, and the less specialized the role, the higher is the likelihood that "if you're above their competency threshold, then there's some team somewhere that needs more people even if they hire someone else for this role".