You need a full 12 volts,and by that I mean 12 volts with some amps behind it..using a multi-meter may show 12 volts,but it may not have enough amps to kick in the solenoid and crank...it needs amps and volts,not just voltage..
I use a headlamp bulb or something similar that puts a good "load" on the wires, to test wires like that one (or an electric fuel pump power wire)...
I would use one to test that wire,and leave the key in the "crank" position for at least 30 seconds,and watch to see if the light dims,or flickers,which would indicate a high resistance somewhere and reduces the available amps at the solenoid..
Part of the problem is the ignition switch gets its power thru wires with fusible links at the solenoid and another link up near the brake booster--after some years worth of corrosion,and the good possibility a short sometime in the past may have partly fried a fusible link,and the distance the voltage needs to travel from the battery to the ignition switch,thru the neutral safety switch,then on to the solenoid,there is a lot of connections and resistance,so by the time it gets to the solenoid,it wont have enough oomph to engage it...but you can jump it at the solenoid and it'll whip right over..
I have by-passed the neutral safety switch on my pickup,and a few other GM vehicles I had,after they proved to be the weakest link in the cranking circuit...that cured the "click but no start" issue,but also makes it possible to crank it up "in gear",which could be dangerous..but less dangerous than lying under the truck trying to jump the solenoid with a screwdriver..
I also installed a push button starter switch wired directly to the battery on more than one truck,using 10 gauge wire...that also was done so I could call it an "anti-theft device" and get a 10% discount on my car insurance..they always whipped right over with that setup..