No. Having a mulch will help with runoff, but compaction won't be enough of an issue to matter.

tj

Walking on soil after a rain compacts it, but the rain itself doesn't do much.

You undoubtedly have some organic matter in your soil. Just spread the mulch or compost on top of the soil. If you have plants, especially new ones, you will need to water a bit more until you can apply the top dressing.

The "fall of water droplets" has nothing to do with compaction. It's about the structure of the soil, and whether there are ways to keep the minerals in the soil from turning into cement. Sand has a no organics, but a structure that largely prevents compaction. A finer grain mineral, such as make up clay, could do it. If you put mulch on top of clay, you'll still have clay down there for a while.

How did you end up with a rare case of little or no organic matter? Roots can bore through just about anything, but organics make it easier for them to do so.

Heavy rain or watering can cause a crust to form on some soils but the underlying structure isn't affected.

Compaction of the soil is the result of pressure that compresses it and eliminates the pore spaces in between the individual soil particles. The finer the soil particle - like clay - the smaller the pore spaces and the less pressure require to compress/compact. Heavy rain or irrigation can lead to compaction (we don't always realize how much force water hitting the soil surface has) but it is usually only a contributing cause and as floral notes, tends to affect only the top couple of inches.

And it is not at all uncommon for topsoils to be very low in organic matter, depending on where one is located.