I wanted to give an update to this in case anyone runs into the same problem.
So I had the alternator replaced ~2-3 weeks ago. A couple of days later, I used my remote start to start the car (this is the OEM Hyundai Remote Start) - it didn't start so I knew something was up again. Now this behavior had been seen before along with the voltage issues that caused me to create this thread.
I had always written this off as the remote start using voltage to determine when the car had started or some fail safe, but the voltage on the battery was good at this point. Then I started to think about this post:
High resistance/open circuit on the alternator output.
What if the Remote Starter was the open circuit? The Remote Starter being a cause didn't even register as a cause for me but the more I thought about it, the timings of when the remote start didn't work and the voltage issues always seemed to match up.
After disconnecting/reconnecting the battery (the reset), I've been mainly just avoiding the remote start and the car has been working perfectly. I still want to cause the issue to happen again so I can find out what exactly on the remote start is to blame (the larger module or the DNA card/thingy) but the weather is getting better so I've been fine without it.
Another thing I noticed when the last issue came about (post-alternator), the voltage was dropping very slowly - like the new alternator had more power to overcome the open circuit.
If I find out what exactly is the culprit, I'll update this thread again but I'm almost positive it was something in the remote start circuitry. Thanks everyone!