The $p$-value is not "an exact estimate of the probability of falsely rejecting a true null hypothesis". This probability is fixed by construction of an $\alpha$-level test. Rather it is an estimate of the probability that other realisations of the experiment are more extreme than the actual realisation. Only if the present realisation belongs to the top $\alpha$ extreme realisations, we reject the null hypothesis.

But it is right that you can imagine the $p$-value to be the minimum $\alpha$, such that the test is on the border of significance to insignificance for the present data.

The $p$-value is not "an exact estimate of the probability of falsely rejecting a true null hypothesis". This probability is fixed by construction of an $\alpha$-level test. Rather it is an estimate of the probability that other realisations of the experiment are more extreme than the actual realisation. Only if the present realisation belongs to the top $\alpha$ extreme realisations, we reject the null hypothesis.

But it is right that you can imagine the $p$-value to be the minimum $\alpha$, such that , if this $\alpha$ had been chosen this way, the test would be on the border of significance to insignificance for the present data.

Maybe a different explanation helps: We say that we reject the null hypothesis, iff the present outcomes can be shown to belong to the extreme $100 \alpha \%$ of possible outcomes, provided the null hypothesis holds. The $p$-value just indicates how extreme our outcomes actually are.