It would be quite a coincidence if the appearance of the wrench light wasn't related to the collision. As far as what's causing it; that's a total crap-shoot. With the first-generation cars, this was a fairly common occurrence, and in the overwhelming majority of cases, the cause was a defective throttle body. Ford issued a TSB, and the defective TBs were replaced. In fact, Ford eventually covered this repair under an extended 10-year/150K mile warranty. You haven't indicated what model year your MKZ is, but 2nd-gen cars have not had this issue.
IDK if you've searched through the forums, but my 2020 MKZ 3.0 has been exhibiting a wrench light on and off since late last year. The car has been to the dealer 3 times and currently, has been there since February 23rd of this year. Despite, contact with Lincoln engineers, to date, they cannot find the root cause. They've run a TSB for stuck VVT solenoids, replaced the BCM, battery cable, a wiring harness and at last update were waiting for another wiring harness. So, who knows? But I don't think that your collision shop is going to be able to clear the warning without intervention from a Lincoln dealer. Is your car still under B2B/powertrain warranty? A dealer may balk at covering this under warranty based on the "damage due to accident" clause, but they may not be able to prove that either. In any case, if this repair was covered under any collision insurance you may have, any costs could be submitted under a supplemental damage claim.
WRT concerns about driving it with the light on, did you have any driveability issues? I did not, and used the car on a daily basis from the time the message first appeared until it went in for service.
Good luck and keep us posted.