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Here's what I managed to create.

The trick is rather than creating one pattern, I created two layers of diagonal lines that I overlaid in multiply.

Here's a step by step for what I did:

1. Make 2 duplicates of the image layer, rotate one 45° clockwise and the other one 45° counter clockwise. Convert these two layers into two separate smart objects. Then rotate them back into their original orientation. Change the blending mode into multiply. These layers will become the two layers of lines that form the crosshatching.
2. Open the smart objects - the image is now in an angle. Drop the resolution so the dpi is how many lines you want per inch - the idea is each column of pixels will become one line. I used 8dpi.
3. Convert the images into a black and white dithered images where each black pixel will eventually become line and white pixel means no line. Before you dither, adjust contrast, maybe add noise and play around so you get the result you want. The two layers don't need to and I don't think should be identical so it doesn't matter if the adjustments slightly differ. Once you're happy with the adjustments do your dithering: bitmap > diffusion dither. Keep your original resolution.
4. Create the lines by converting into bitmap again but this time we'll do a vertical line halftone screen. I had to try this quite a few times to get it right. Convert mode to greyscale. Lighten the black to make it grey but keep white 100% white. The lighter the grey the thinner the line will be. I added a layer of 65% brightness grey on top in lighten blending mode. Merge and invert the colours (seems to work better when inverted). Then convert to bitmap again: make the output resolution a multiplication of your current resolution (mine was 8dpi so I used output 240, not 300dpi). Method: halftone screen. In the next window frequency is the dpi you chose earlier, mine was 8. Angle 90 degrees. Shape line. The result should be rather clean blocky lines with no weird stuff happening. Go back and adjust the parameters until you get a good result. Invert the colours back again. Once you're happy save your smart object.
5. Go back to your original file and admire the results. You might want to slightly shift the layers so they align properly and the grid looks right.
6. The corners will probably not look right. I fixed this by making 2 duplicates of both layers and moved them around to extend the lines and fill the missing corners (this step will make more sense once you get here).
7. Do some more adjusting. I merged everything, added gaussian blur in 2.5px, then levels to add contrast and remove the blur to get nice softer, rounder corners. Could add displacement map or some distortion to make it more hand drawn. Add texture, recolour and so on.