I guess it's drawn. It can originally be a blurry, but half-toned letter "a" which is zoomed to bigger size, treated like you have already done to get the bridges between the dots. The not so perfectly circular dots are changed manually to regular ones.
The final pattern can be tiled in Illustrator. The used atoms seem to be these:
Circles (A) and squares are combined with the pathfinder boolean operations to shapes B, C and D. In this phase I avoided the shape builder because it often changes the anchor points or generates more of them. Exact snapping becomes difficult or it doesn't happen at all if the anchors are redistributed. Pathfinder booleans have always kept the original anchors if the parts are placed perfectly so that the anchors snap.
Parts A, B,C and D can be tiled freely. The parts snap perfectly if one has snap to point and smart guides ON.
Finally the areas can be filled with the shape builder. The same tool can be used also to make holes. Pattern E is a simple example.
Not tested, but one could as well tile the plane to full of circles and then use the shape builder to combine the wanted ones and to remove the extras. Then some rectangles can be inserted for straight edged parts.
After tiling the artwork has got colors, blur and grainy texture. That's easiest in Photoshop.
