I hear an occasional loud vibration coming from the front end, driver’s side of my 2003 Honda Pilot. It sounds like I’m driving over the rumble stips in front of a toll booth. When I take my foot off the gas, it goes away. It’s been happening for a while now, my mechanic thought it was loose screws, but tightening/replacing hasn’t helped. The car has 150k miles on it. The rumbling has been going on for about 50k miles.
Foot on the gas vibrations are often caused by bad CV joints which if your pilot is AWD has in the front. The joint boots may look fine, but the joint can still be bad. Grab the axle and give it a good shake vertically and if it moves more than say a millimeter it’s likely the problem.
I agree, likely a bad CV joint - I’m surprised your mechanic didn’t think of this, it’s the first thing I’d think of with your Pilot’s symptoms.
On the Acura MDX, which is mechanically similar, many people describe their failing torque converters as having that exact noise. Search for “rumble strip” at http://www.mdxers.org/forums for more details.