Right off the air handler, there's a big rectangular duct and then there are other flex ducts off of that to feed the rooms.
That is not going to help your situation. If flex duct is used all the way from the furnace/air handler to the diffusers, that's going to introduce a fair amount of pressure drop. Flex duct is not smooth on the inside, and is therefore fairly resistant to airflow. It's best used as a replacement for elbows where it's difficult to get an elbow to line up well, but it's often used as the primary duct because it's easy.
Seeing as it's bigger it will have less resistance than the smaller ducts and would get more air. Is this a safe assumption?
Well, not exactly. The resistance is going to be comprised of the resistance along the entire path. The resistance after the box, where individual 6" duct goes from the "box" to the individual diffusers is going to dominate the total resistance for the run. Just because the duct between the "box" and the initial plenum off the air handler is larger doesn't mean it has less resistance. It's also handling more air (the sum of all the smaller runs downstream of it). It's the velocity of the air that's important.
I have no dampers anywhere in the system.
That does not surprise me. In residential systems, I think most HVAC contractors consider the dampers in the diffusers to be the balancing devices, even though technically that's not what they are
supposed to be.
Last night I partially closed a lot of the registers downstairs to see if it made a difference upstairs but it didn't really do much at all and made a lot of noise.
Then that tells you what to expect from zoning dampers. The only difference is: they'd be automated, and would likely make less noise since they most likely wouldn't be right at the outlets. But from a temperature balancing perspective, closing off the diffusers, or having a zone damper that closes, is going to provide the same result. If the former didn't help, neither will the latter.
As I mentioned earlier, it's really hard to get a good balance in a two story home with one system. If you don't have access to make significant changes to the duct system, you probably won't get really good results without splitting the system up, or adding spot cooling from something like minisplits where needed.