As you mentioned, there are both "transitive" and "intransitive" 触れる (What is the difference between に触れる and を触れる?). The transitive 触れる is for touching as a result of somebody moving their body, like your example 肩に手を触れる. It describes an action under one's control, thus usually used when touching or non-touching is most meaningful in that movement. On the other hand, the intransitive one (e.g. 肩に手が触れる) relatively focuses more on the state being contacting in the action.
So, there are following possibilities why the writer preferred 触れさせる:
- to emphasize the fact of "keeping/maintaining touching" (or perhaps "continuously tapping" in your first example)
- to say "let them touch", which suggests the contact is fairly light, like the hand barely feels the skin
- just the writer's personal habit
Incidentally, as far as I image those situations correctly, I don't feel it comfortable to use the transitive 触れる either. Instead,