***SOLVED***
First off, I would like to thank you all for your help. Between all of your insights I was able to figure out the problem.
LONG story short, I hooked a 9V battery to some speaker wire and started poking my fully assembled cluster (now on my kitchen counter) until lights started turning on. To my surprise, sometimes almost all of the lights turned on! WEIRD! After a while I noticed a pattern that when I put voltage across each light socket, one of two things would happen: 1) Only two lights would light up OR 2) All of the other lights besides two would light up. This would happen after changing the polarity of my test by reversing which side on each socket I had the "positive" and "negative" cables from the 9V battery.
I then checked the forum to post my findings and saw what @danno had said about the insulating brackets so I took apart the cluster again and made sure they were in the correct position. Once I re-assembled the cluster and repeated my earlier test, I got the same results EXCEPT now two different bulbs were lit up (or not lit up depending on the cable polarity).
Once I bashed my head against a wall a few more times, I thought back to what @Midlife said about polarity, and even though the bulbs I purchased have reversible polarity, and, therefore, should not care how they are installed, I thought maybe, just maybe, technology back in 1969 wasn't set up to handle reversible polarity.
I then, in the proper "positive/negative voltage flow" scenario, removed and rotated each bulb, that wasn't lighting up, 180 degrees and BOOM. All bulbs work as intended. What a relief.
TLDR: Even reversible polarity LED bulbs bought from modern manufactures are not reversible in this context. I have no idea why and random chance of how I chose to install each bulb has been my downfall all along. I will definitely be marking my lights and light sockets with a pen from here on out.
Anyways, a test drive to make sure my gauges were working revealed that the only casualty is my fuel gauge, which does not work anymore. @danno this gauge had the insulating brackets in the correct place all along so i'm not too sure why it's suddenly not working, but I am taking the win for tonight.
Pictures attached of my 9V test for those who are curious. I hope this thread helps others who have issues with after market LED's.