I've cleaned the finish switches and checked to see that they're adjusted properly. I checked continuity of the wiring from the switches to the connector.

When horse numbers 1, 3, 6, and occasionally 2 reach the end, it doesn't appear that is being detected - the respective lights don't come on and I believe the star for the odd/even side is supposed to advance.

Nothing happens when I manually contact any of the switches, even the working ones.

There is continuity on both sides of all those switches when they are not touching.

I followed the schematic and inspected everything that makes connection from the finish switches to the relays to the score motor. Of course I may have missed something. Everything else seems to be working correctly on the game. I lack understanding of the theory of operation of that horse track mechanism. I'm stuck on what to look at next. Any ideas would be awesome.

Good idea, thanks. It's at a friends house and I haven't had a chance to check that yet. However, even if there is too much resistance on some of those connections, I'm baffled about why nothing happens when I manually trigger the working switches.

Instead of looking for a connection between both sides of the switch, I should be looking for voltage? Would using one of those light bulb testers work for something like this?

Follow up:
It turned out to be dirty switches. Originally I "cleaned" them with a piece of paper, the only thing handy. This time I had my tool bag with my trusty flex stone file.

The #1 Relay and #1 Release Relay was working so it had to be a problem with the #1 Switch. Unfortunately, I confused that with the one labeled #1 Back SW. I took that to mean the switch in the backbox. Then I realized there are targets at the back of the playfield mounted under where the backbox sits. I had someone push that target in while I shorted the finish switch and presto! I cleaned the switches properly and it's all good now. Lordy.