I can confirm that the problem seemed to be that the 6A PSU was too weak. With 8A and 10A PSUs it worked fine.Thank you!
I don't have any more powerful PSU, but I have ordered a 10A one arriving later this week.
I also ordered a power usage meter.
I'll do some experimentation and report back.
Previously I have tested the Macbook power draw with a USB power meter, with the following results (when the Macbook battery is on low charge):
- Only plugging in USB-C: Macbook draws ~80W.
 - Plugging in both Displayport and USB-C: Macbook draws ~40W. The USB-C connection still dies if I put the display on high brightness in this scenario.
 - Plugging in Displayport and then USB-C through a dongle: Macbook draws ~20W. This was stable with high brightness if I recall correctly.
 
I did some measurements to check the power draw from the outlet, the peak I saw with USB-C charging on low battery + 100% display brightness was around 165W. So if you want to use the 90W charging over USB-C you need at least an 8A / 24V PSU.
On full battery charge the draw over USB-C was around 30W with 0% brightness, and 75W with 100% brightness.
With Displayport (meaning the display is not powering the computer) the draw was around 20W with 0% brightness, and 65W with 100% brightness.
Maybe this can help someone else too!