A bit implausible for the instructor to throw that in - if your within 1 or 2 minutes of landing, hopefully ATC won't be lining up traffic ahead of you.
I'm with you - I would have continued to land, in much the same vein as if the Captain had a heart attack at 6 miles I too would continue, the mentality being the sooner you are on the ground the better! Just call the Mayday with ATC and TELL THEM what you are doing.