I have the solution! I feel a little stupid because it could have been a 0 dollar fix. After recording the video, it quickly got to the point where I could not start the car at all, so a voltage drop test could not be performed. I think it is done with a running engine and under load. A ground point was the culprit, most likely the one between the engine block and the body. It seemed fine, but under the screw it was corroded and there was no good contact. I went through all ground points that I could reach, also sand down the battery terminals and it fired right up. The burning smell really came from the solenoid, because of the repeated tries when it was clicking I think it just overheated.

I should not have replaced the battery and the starter, but the battery was 6 years already (came with the car) so it may have needed replacement anyway.

Thank you for all your help, and ideas.

 

Hi five! You fixed it!

 

Congrats on finding the fix.
Thanks for the update.

Corrosion the number one killer of nice cars ... imho ... lol

 

I just wanted to post a link to that I followed and it worked- this was after looking at dozens and dozens of threads and trying just about everything. The link is: https://www.hyundai-forums.com/lc-20...-solution.html
It's worth a try and I hope it helps someone out there!