"Let's" And "Let Me"

From my grammar book

Pragmatically let's is three ways ambiguous: (i) you and me, (ii) you alone, and (iii) me alone. The last two readings are illustrated in the following:

a. (mommy to kid): Johnnie, Let's wash hands.

b. (daddy to kids): Everyone, be quiet. Let's hear the phone message.

In (a), the mother and her child both are involved, if she somehow helps him/her to wash hands. On the other ahdns, if the mother is just suggesting her child to wash hands, only the child is involved, which is based on Grice's (1975) cooperative principle. In the context of (b), only the father is involved, which is based on Grice's cooperative principle.

In (b), according to the book, if only the father is involved, why doesn't he say "Let me hear the phone message" in the first place? Why does he say "Let's", not "Let me"? I think in (b), let me is more intelligible than let's.