Been working on the '72 Honda like crazy lately. Here's an update.
After all my fiddling about with the fuel system, nothing seemed to be working. I pulled each spark plug to see if fuel was even making it into the engine, and two plugs were dry after turning the engine a few times over. This indicated to me that the middle two carburetors were having troubles, so I decided to pull the carburetors and just thoroughly clean and rebuild each one.
Pulling the carburetor assembly out of the bike was a bit of a chore. I don't think they've ever been removed completely since it rolled off the assembly line! I removed the throttle cables and spent about twenty minutes tugging on the assembly before it loosened and I was able to completely remove it. Then the task of separating the four carburetors, which was a bit more complex than I thought it would be but I managed it just fine. Now I had four carburetors and a small pile of linkages that synchronized their throttles, chokes, and fuel lines together.
Taking apart the first carburetor I could've sworn I lost a spring, a tiny little compression spring (there's about a bazillion little tiny pieces to each of those linkages, including little tiny springs). I spent about an hour searching the floor of my garage, stopped for the day, then took a closer look at the exploded diagram of the carburetor and it turns out the spring I was looking for was still inside the linkage. Lucky break. Actually it was only when I'd dismantled a second carburetor that I discovered that the spring I saw in the diagram was further below where I thought I'd lost one. So I continued the rebuild.
It's becoming apparent to me that this is a lot harder to do without a full set of tools. When I say a "full" set, I don't mean one of those hundred-or-so-piece tool kits you see at the Home Depot, even the ones labeled "Mechanic's Tool Set". I mean a full set, including taps, dies, punches, cold-metal chisels, every single metric combination wrench in the damn world, screwdrivers of every possible size... a full set. Someday I'll have that. For now I'm using a sewing pin as a centering punch, because even if I bothered to get a punch set I still wouldn't have anywhere to put it when I'm not using any punches.
Finally got the whole thing dismantled and found the problem immediately. Either one or both jets of each carburetor are completely clogged, at least a couple of them so badly that they're still soaking in carburetor cleaning solution in my garage right now. I'll try them again later today. The rebuild part is easy enough; I've just been replacing each O-ring and gasket with my spares when I see them. Hopefully once this process is done, not only will the bike actually start but those carburetors will last another few years at least without failing or leaking.
My plan is to finish the rebuild, get the carburetors back on the bike, get it running, then take it to my mechanic, Chewy, to have him synchronize them and adjust them back to where they should be, and replace the tires and a couple of other things on the bike in the process. Once that's done, the bike will be driveable and should be pretty reliable. At that point, hopefully the transmission and clutch and all that I haven't had a chance to try out because the bike doesn't run, will just work and be fine, otherwise I've got a long road ahead of me with this.
So wish me luck, space cadets.