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It depends on the configuration of true group means and on the specific post-hoc tests you plan to conduct.

To make anova and posthoc tests comparable, you need to be conducting posthoc tests for at least $G-1$ contrasts (where $G$ is the number of groups) such that the contrasts span the space of all possible contrasts. In that case, accepting all the contrast null hypotheses implies that the true group means are all equal, equivalent to the anova F-test null hypothesis.

Given this assumption, which is more powerful depends on the configuration of true group means and on the specific post-hoc tests you plan to conduct.

Approaches 2 and 3 test a number of null hypotheses besides the overall null of no differences, but I am answering your question in terms of the overall null. For approaches 2 and 3, the minimum adjusted p-value is an effective test of the overall null because the union of the null hypotheses for the individual t-tests is equal to the overall null hypothesis.

Approaches 2 and 3 test a number of null hypotheses besides the overall null of no differences, but I am answering your question in terms of the overall null. For approaches 2 and 3, the minimum adjusted p-value is an effective test of the overall null because the intersection of the null hypotheses for the individual t-tests is equal to the overall null hypothesis and the union of t-test alternative hypotheses is equal to the F-test alternative.

Approaches 2 and 3 test a number of null hypotheses besides the overall null of no differences, but I will answer you question in terms of the overall null. For approaches 2 and 3, the minimum adjusted p-value can be viewed as testing the overall null because the union of the null hypotheses for the individual t-tests is equal to the overall null.

The F-statistic can be viewed as being equal to a weighted average of the squared t-statistics from approaches 2 or 3. Hence it works best when all or most of the t-tests contribute meaningfully to the average. For the t-test approaches, the result is driven mainly or entirely by the largest t-statistic. In general, the F-statistic will give a smaller p-value if the individual t-statistics are all similar in size whereas approach 3 will give a smaller p-value if one of the t-statistics is much larger than the others in absolute size.

Approaches 2 and 3 test a number of null hypotheses besides the overall null of no differences, but I am answering your question in terms of the overall null. For approaches 2 and 3, the minimum adjusted p-value is an effective test of the overall null because the union of the null hypotheses for the individual t-tests is equal to the overall null hypothesis.

The F-statistic can be written as a weighted average of the squared t-statistics from approaches 2 or 3. Hence it works best when all or most of the t-tests contribute meaningfully to the average. For the t-test approaches, the result is driven mainly or entirely by the largest t-statistic. In general, the F-statistic will give a smaller p-value if the individual t-statistics are all similar in size whereas approach 3 will give a smaller p-value if one of the t-statistics is much larger than the others in absolute size.