I believe what Honda (via custom Omron part) has done is just as you describe. My understanding is such treatment is common when environmentally sealing a part. When I considered making my own little PCB w/ a PIC micro and some relays + supporting components, I was going to glob it with non-conductive potting compound as well. I don't prescribe any evil plot by Honda making the part as described in order to make it difficult for DIY types to fix. After all, how many people would even figure out the problem to an extent where they pin pointing the defective component let alone break open a sealed part in order to save a relatively small amount of money. Most people would just let the servicing dealer fix it. I went beyond that as I've started to consider the local Honda power equipment dealers either somewhat clueless or WAY overpriced for the services they provide. I left it with an authorized dealer over last winter so they could poke at this problem. They actually suggested I scrap the unit and buy something new because if it was the relay the part alone was around $300 or more before labor. I compare the cost vs. having to use a lawn service instead of doing it myself. $125 is pretty inexpensive in comparison! I think Honda was just using good engineering practices. It is an expensive part, but they also have costs associated with maintaining an extensive parts inventory on equipment they haven't sold for a very long time. Also, as a custom part, their costs were no doubt much higher than stock components (i.e. custom tooling, etc). You could argue the equipment might be a bit overbuilt, but then you're approaching Home Depot type disposable power equipment rather than something built to last a very long time (mine is 19 years old and can see it going double that).
Regarding the electrical diagram... At least it's included in the manual. On the small side, and not in color, but still usable. You could scan it, enlarge it and maybe even color code signal lines with markers to make it easier to use. I've seen component data sheets (e.g. Bosch accelerometers) which are truly disasters and horribly difficult to use. I wouldn't expect to see sub-components (e.g. combination relay) detailed in any public documentation. Wouldn't be at all difficult to map out the various states of the combination relay and program an inexpensive ($2 - $3) microcontroller. Once mapped out, you could probably do it without a micro - didn't think it through to that level. I considered doing it, but there are actually much more interesting projects I'd like to tackle, so rejected that approach. Also, my lawn continues to grow, and wasn't waiting for me to come up with an alternative. Luckily, my neighbor mowed my lawn with his 30+ year old John Deere while I waited for the part to arrive.
Sorry - just realized you're trying to use the on-line manual rather than the paper copy one. Haven't looked at it, but could believe it's not great resolution. If it would help, I'd be happy to scan the one from my paper manual at high resolution and email it to you. Let me know.