A multimeter isn't going to tell you whether the battery is good or not, all you'll see is 12.8 or so for voltage. You can still see 12 volts on a bad battery. New battery doesn't guarantee a good battery. Needs to be fully charged and then load tested. If nothing else, pop the battery in someone elses bike and see if it works on their bike.
One thing you can do is try to spin it over with the plugs removed. It's kind of shade tree but if it spins over with the plugs out but not with the plugs in the battery may be the culprit.
Next do as you plan and clean all the connections down to shiny metal. Battery terminals, starter terminals and all grounds need to be clean. Hook them back up. I always use dielectric grease to help cut down on corrosion.
If the cables are original you're looking at 25 year old cables. They can build up corrosion down the length of the strands in the cable itself. This creates resistance and results in hard/no starting, even with nice clean connections. So, even though the connections are clean you can't overlook the cable itself.
Even if you had a short somewhere else the engine should still spin up fine, provided the battery was good and your cables/connections are good. If you have gremlins in another circuit it may not start, but as far as spinning over nice and strong, it should, if all is well with the starting circuit components.
Hold the cables when you try to start it. Does it feel like they're getting warm?
If it were me, I'd be focussing on the battery and cables first. Other than that, worst case scenario is a bad starter windings that may have shorted out over time. For spin up issues there's really not much more than the battery, cables, and starter involved. Now, if it spins up nice with a good battery, and 3 days later it won't start then I'd be looking for a drain somewhere. But with a fresh battery....good cables and connections....and a good starter, it should be turning over.
If you still feel you have an issue somewhere else pull all the fuses except any affecting the starting circuit, and start from there.