Bill, thank you for posting the pictures, and Adam, thank you for taking them.
I'm with mechanix on this issue. The view through the oil fill hole demonstrates that there is varnish. It doesn't look like excessive varnish, but it's there. Is it worse further inside the head? We don't know right now. Did the technician at the dealership remark about an excessive level of sludge or varnish formation? I imagine he removed that front cylinder head cover to adjust at least the front valves. If the technician found the cylinder head to be caked up as much as some of the stuff that Trav posted, I would like to think that he would have mentioned that.
His (technician's) observation is that the cams were worn down too much to adjust the valves. There's a wide range of valve adjustment available, and I'm not sure I'd take his observation as gospel at this point. In fact, if you recall, I've been skeptical of the cam wear philosophy from the beginning. Fuel economy is the same as before, power is the same as before, and you notice no other symptoms besides the loud valvetrain. Is that correct? If the cams were worn too low to adjust properly, I have to think that you would feel a significant reduction in power, or there would be other operational symptoms besides only a clicking noise.
Here's what I would do. If I were mechanically-inclined, I'd remove the front cylinder head cover myself and check the valve lash adjustment. This requires some knowledge of the operation and at least a set of feeler gauges. If you're not comfortable with that, I would have an independent garage diagnose the problem. I wouldn't let them "fix" anything less they called you first. They may say that the cams are fine and all they need are an adjustment. They may confirm the technician's observation that they're ground down too much to adjust right. At this point, we don't know.
So therefore, I will also parrot what Trav has already said: get a second opinion, and I wouldn't get a second opinion from a Honda dealer. I'd get one from an independent garage that you trust or that comes recommended by others. Someone else taking the cylinder head cover off to measure the cam wear will in no way, shape, or form change your chances of Honda helping you out on this if there is indeed a mechanical engine problem. But it will confirm what's already been told to you (or disprove it).