We towed a popup with a Honda Ridgeline. Did okay - not great but okay. We bought a Jay Feather X213 @ 5500 lb max and never even considered using the Honda. Traded for an F-150 3.5 EB 4x4 w/tow and never looked back. I would no more consider towing my TT with the Honda than I would tow a 40 foot fiver with the F-150. (My understanding is that the Ridgline is not supposed to be used with a WDH, and I wouldn't tow a regular TT without one.
A couple of months ago I was passing an Odyssey on the road and happened to see that it had a hitch receiver... looked tiny, about 1", like it might tow a little red wagon but not much more. Even the Ridgeline set up for towing came with a 2" receiver.
One last comment. The F-150 that the OP is comparing with the Odyssey has to be the absolute bottom end of Ford's line. Mine is not at, but closer to the top end, and there is a lot of room in between for various options and body styles. My F-150 could tow my TT plus his Odyssey and not exceed tow rating. it wouldn't be a lot of fun, but it would do it. I'd make the same comment about towing any actual TT with the Odyssey... some might be small enough for it to manage to pull them, but it would not be a pleasant experience.
I towed my Jayco back to Colorado yesterday after 5 days camping in Nebraska and Iowa. We cut the trip short because we were tired of rain and thunderstorms and the next 2 days still forecast more of the same. We drove 571 miles in 9¾ hours, most of the way on I-80, in rain and strong, gusty winds all the way. I put the cruise on 65 and drove a mix of one and two handed all the way with no problems, even when being buffeted by passing big rigs. That would have been a nightmarish drive, even towing our popup with the Ridgline. My properly set up F-150 is just that much more capable.