Why Hypothesis testing emphasizes the rejection

Question

When I take a statistical inference course, the professor said that hypothesis testing emphasizes the rejection, and usually we would say that we cannot reject $H_0$ than accept $H_0$.

I'm confused about this statement and interpretation of hypothesis testing result.

Denote null hypothesis $H_0$ VS alternative hypothesis $H_a$, and test method has type 1 error $\alpha$ and type 2 error $\beta$.

Don't the two parameters mean:

1. the probability of correctly accepting $H_0$ is $1-\alpha$
2. the probability of correctly accepting $H_a$ is $1-\beta$

or equivalently

1. the probability of correctly rejecting $H_a$ is $1-\alpha$
2. the probability of correctly rejecting $H_0$ is $1-\beta$

Then why we don't just interpret that

1. if we accept $H_0$, we have $100 \times (1-\alpha)$% confidence that accepting $H_0$ is the right choice.
2. if we reject $H_0$, we have $100 \times (1-\beta)$% confidence that rejecting $H_0$ is the right choice.

So how can hypothesis testing emphasizes the rejection, and say that we cannot reject $H_0$ ?