Chain, it won't help, just the opposite. I guess I wasn't clear on that. Basically as you've experienced, anything that lowers the front will worsen headshake. Also less pressure gives a bigger contact patch which can be more sensitive to the pavement.
TBD, the bearings break-in, and Honda puts almost no grease in them from the factory. Most people here have dampeners so they don't really notice it (or just add clicks without thinking why), but I'll bet you at least 50% of the bikes on this board have headbearings that are in need of grease and adjustment.
When you buy a new bike, if you really care about performance, you should take it for a couple of rides and then basically pull off the front and rear suspensions to lube and retorque all of the pivots and bearings. I know it's a lot of work but it is time well spent. If you buy a used one, you should do this right off the bat.