Why do depleted blood veins seem to appear near the surface (or skin) more frequently than blood vessels and arteries?
Short Answer:

The veins you see are special veins right at the surface; major arteries are typically much deeper in the tissue.

Longer Answer:

Great question. Usually veins and arteries come in more or less one-to-one pairings, traveling side-by-side (often nerves accompany them as well). Both types of vessels tend to be fairly deep, because, as you note, damage to blood vessels can be quite problematic.

However, superficial veins, the ones you can typically see right under the surface of the skin, are different. They don't have a direct artery pairing, and they are not the only venous drainage from the areas distal to them.

Instead, superficial veins have a known purpose in temperature regulation. When core temperature is high, the superficial veins are relatively dilated to allow temperature exchange between the blood and outside. When core temperature is low, the superficial veins constrict to cause the blood to flow deeper and retain heat.

A nice side benefit to these superficial veins is that, because they aren't completely necessary for circulation, they can be harvested as surrogate vessels to divert around a blocked artery, like in coronary artery bypass grafting. The saphenous vein from the leg is often used. Superficial veins can also be removed for cosmetic reasons.



References

Webb-Peploe, M. M., &amp; Shepherd, J. T. (1968). Responses of the superficial limb veins of the dog to changes in temperature. Circulation research, 22(6), 737-746.