The core issue is that p-values are cheaper to get than replicating the study, but replicating the study is the only reliable way to see if it's true or not. Sometimes the expensive/time-consuming way, is the only good way.
The tables are misleading too: of course p(h0=T|p<.05) is going to be about 0.50 if your power is 0.50 and p(h1) = 0.10. That is maybe the bigger problem than p-values per se. Also, he fails to compute the other cell, which is p(h0=T|p>.05), which ends up being about 0.95. The problem in that table is have no power to detect anything.
In real life, you don't know p(h0)--that's the point of doing the study. So Bayesianism doesn't really help you.
The real problem is lack of power.