There is a cool trick inside the tile sampler node that you can try. If you make a normal map and use it as a vector map to rotate and orient the samples.
Like this:

After doing what you have shown, I still can't manage to get the tile sampler to give me that output no matter what I try.

EDIT:
As usual, not 10 minutes after posting this, I fixed my issue.

Using color mask map threshold set to 1 and then adjusting some other settings, I got the output I was looking for. Thank you to everyone that helped and this topic can be marked "Solved" now.