I am experiencing this a very similar problem in a major project that I am working on, ever since updating to Logic 10.5 (OS 10.14.6).
It typically works for a few minutes, then I have to quit/reopen logic for it to start playing again. Alternatively, if I record audio, it plays back from the playhead just fine. If I simply push play, it will only play for split second, then stop again. This is very annoying as I don't want to be recording audio all the time just to be able to hear what I am doing.
Unlike the OP, I cannot play the project from the beginning - it's either record or nothing.