Edit: I am taking a break now and will return later this afternoon/evening to respond to any comments and answer any questions. Thank you everyone for a great and engaged AMA so far.
TN status is conceivably something you could get without help from the employer (though even the smallest startup hired one to help me set up a package). What you need is basically proof that a company wants to hire you (offer letter), evidence that the company exists, and proof that you fall in one of the TN occupations.
I've been through this process many times, and I would always say yes and then mention with the recruiter (even if I had to explain that the process was basically "asking politely for permission at the border").
The underlying question that employers really care about is "will you be legally allowed to accept a job offer without unexpected expenses or delays". Even if you file through USCIS (and not petition at the point of entry), you can (should) have an answer in 14 days and ~2k in fees, which is 1) a drop in the bucket for any hiring budget, and 2) not impactful to a hiring timeline.
No portion of my initial comment should have been interpreted to mean that employers do not do their own due diligence after actually hiring someone.