In sentence preceeding the one you've quoted, the author refers to Eisntein's laws of general relativity and the Standard Model, and with that context the sente

In sentence preceeding the one you've quoted, the author refers to Eisntein's laws of general relativity and the Standard Model, and with that context the sentence makes sense. These 'classical' models do indeed suggest a time-symmetric universe. The whole problem here is reconciling classical models with quantum mechanical models, which as you've noted do not suggest a time symmetric universe.