There is whole group of people in social sciences who are pushing for abandoning the null hypothesis testing methods. For the reason you mentioned, I cannot believe changing the test (or threshold of the test) would solve this issue. You ask people to find something significant or fit a model to a data — and tell them that's what matters — and they'll do it, either intentionally or unintentionally.
Machine Learning will (have) the same issue if the only thing that matters is hitting a certain level of accuracy given your model and data. This has been observed in Kaggle competitions over and over, you ask a group of people to find the best fit, and they'll, by learning your train, validation and test datasets.
As mentioned, problem is not p-value, or null hypothesis testing, the problem is journals who promoted the wrong incentive, and educators who were not aware of the consequences and propagated the wrong incentive (interpretation) to students.