I do recall several other people posting about problems with Odyssey motor mounts over the past couple of years. That recollection is not a scientific study regarding the possibility of a wide-spread problem with that part, but the “dealer’s” statement does seem to jive with my recollection.
Unless Honda of America decides to help you with a “good will” warranty on this part, then you have little choice but to pay for the repair yourself. It wouldn’t hurt to contact Honda of America’s Customer Service staff and politely ask for monetary assistance with this repair. If you have consistently had the car serviced by the dealership, then you have a better chance of some assistance.