Ecobee restarts when calling AC - old analog thermostat worked
I talked to my landlord to inquire about access to the actual equipment and it turns out its on the roof and not happening. However, he said he had the same problem with his unit and his Nest and to solve it he just removed the C wire. A big difference between the Ecobee and the Nest is that Nest has a battery and therefore does not require a C wire. I got one, wired it up with the same config as posted, minus C wire, and it was able to call the AC right away. Some commenters mentioned that my C wire might not actually do what it is labeled as, but it did power on the Ecobee. So either I had a bad Ecobee or it isn't able to handle certain funky situations that the Nest can. Maybe the lack of C wire affects the current draw, but hard to say since I never accessed the equipment.