If the holes or tunnels are really only 1 inch in diameter, I'm guessing wasps or bees. Try watching the holes for a while and see if anything flies in or out. I have had them before and sprayed wasp & hornet spray into the hole, saturating the area, them filled the hole with soil and saturated the soil with more spray. Later on walk by and if you see bees buzzing around where the hole was, spray them, too. It could also be chipmunks, but their holes are more like 2 - 2.5 inch diameter. I've got them now, too. You can keep filling the holes, but they keep digging them out again. You have to trap & release them, or humanely destoy them, first.

No, increasing the particle size by adding coarse sand helps tremendously and is essentially permanent. You're trying to create space in the soil where fungi can grow and water and air can penetrate. Organic matter gets used up, it's eaten by the soil bacteria and worms which convert it to substances that plants use. You have to keep replacing it. Do both, but obviously the organic matter is more important. Does it take a lot of coarse sand? Well, sure it does. If you have hundreds of tons of clay soil in your yard, one sack of sand isn't going to do much. You're trying to change the proportion of large particles (sand) to small ones (clay).

Rather hard to see but if there is a mound around the hole made up of small balls then they were made by land crayfish. They live in moist/wet soil. They feed on soil invertebrates. In MI we would find them after long heavy rains. Not a problem.

I've been hearing cicadas in my neck of the woods in recent days. Sounds like the holes made by emerging cicada nymphs. Their tunnels can be very deep. Several kinds of solitary bees and wasps make 'single unit' nests in the ground. Have you observed any such activity? You made no mention of little burrows in your yard, so I'll guess that it's not mole crickets.