Steering wheel vibration when braking with warm brakes a mystery! on 2003 Honda Odyssey

Bought a 2003 Ody with around 150K miles. It drove like new except for a vibration when braking, which I figured was balance or rotors worst case. Upon driving more I discovered the vibration happens only when braking and has no accompanying pull even when cornering. It is only in the steering wheel, the passengers do not feel it. When the brakes are cool - first drive in the morning, when it is raining and on cool evenings - the vibration is minimal or gone. I do not know the history but it was advertised as having been significantly rebuilt. It appears to have a like new front end. Half shafts and struts look nearly new, alignment was perfect - not just good. There is zero play in the steering while driving and when on the rack nothing is even a little loose or worn that we can see - just like new. It had mixed age Michelin Harmonies and decent brake pads. We decided to replace all the pads and front rotors and noticed that the lower caliper pin with the rubber was replaced with a straight one on the right side. Replaced the pins with new OEM and the vibration was still there. Replaced the rotors again and we were careful to check for dirt and torque the wheels, but no avail. I replaced the tires with 4 new Defenders, and still no effect. I recently took it on a trip loaded down and had no issues, in fact the vibration seemed to have been less pronounced fully loaded. I am out of ideas. Any one experienced and resolved a similar issue?

Asked by for the 2003 Honda Odyssey
Check tie rod ends and ball joints, also check control arm bushings..
Thanks TBone. There is zero play in the wheel and nothing loose underneath after close examination on the rack. The shake only occurs when braking and increases with the intensity of the braking. If the brakes are kept cool as in rainy day and when starting out in the morning, the problem does not exist. Don't see how it could be a suspension part. Am I wrong? Rotors have been replaced twice, pads and pins. Same result with both sets of new rotors, and same as with the old brakes. Any other ideas?
when braking your applying prewssure in one direction while the rest of the vehicle is still moiving foward
the force from the brakes pushes the spindles backwards or toward the rear of the car while the rest is still moving foward if a bushing is worn it would allow some degree of play and creat a vibration.
also may want to look at upper strut mounts, is possible that they have went bad...
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I have the same issue with my 2008 Odyssey w 104k miles. I was told it was the way the rotors were shaved before, and it wasn't a safety I ignored it. It seems to be getting worse, so if you find out how to fix it, please post! Appreciate it.