Sooo, just as Chewy4u before me, I solved the issue with Deda compression ring, which I bought here:
https://dedaelementi.com/compression-ring-dcr-headsetThe shipping was 3x the cost of the ring, but it did work. Well, let's not get ahead of myself, I did not test it outside yet, but the obvious play I got with the original C-ring is totally gone.
What I did with the Deda C-ring is removed the amount of material from the top to match the original C-ring in height, and then just a little bit more (less than 0.5mm). You can test this if you lay them side by side on the table and then they should be the same height. I wish I had taken a little bit more material since I still have a small gap between the frame and the headset cover. I have not used any additional microspacers.
I tried to remove the excess material myself with the saw and files but it was hard and I could not get it to be perfectly straight. I just took it to a local key maker shop that had a spinning grinding tool - it took him 10 minutes to finish.
I think what Chewy4u said makes sense - the original C-ring just has too small of a radius which does not fix the fork steerer tube in place. This is a fault by Velobuild and they should fix this I think. I lost almost 3 weeks fixing this in which I could not ride my new bike. Hope Chris from Velobuild sees this and makes sure to fix it for the next buyers.