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I still do not follow why here they say and where does the asymmetry of

Pr[reject $H_0$$H_0$ is true]

and

Pr[accept $H_0$$H_0$ is true] come from.

And they continue:" The $p$-value is a conditional probability where the condition that is assumed is the truth of $H_0$. It is calculated under this assumption; therefore, it makes no sense to ask

Pr[accept $H_0$$H_0$ is true]

because if you presume the truth of $H_0$ then the test statistic for whether to accept $H_0$ will not be a random variable--it will be fixed because you fixed it in your assumption."

I think that we also fix our assumption if we say symmetrically

Pr[reject $H_0$$H_0$ is true]

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