Confusion about the Bellman Equation
$p(s', r | s, a)$ is the probability of arriving at state $s'$ and obtain reward $r$  given that the environment was in state $s$ and the agent took action $a$. Therefore, this probability is defined assuming action $a$ is taken. There is no probability of taking $a$ included there.
The probability of the agent taking an action is provided by the policy $\pi$, and that is why we need it in the equation.
You can think of the interaction of these two terms with the law of total probability: $p(A)=\sum _{n}p(A\mid B_{n})p(B_{n})$, where $p(B_{n})$ is analogous to $\pi(a|s)$ and $p(A\mid B_{n})$ is analogous to $p(s', r | s, a)$.