The issue is that when we talk about what the Lord Buddha taught we should really consider how He or His early students used a specific word in the whole corpus of the Canonical texts , we should analyze the entire context. Neither your opinion, nor mine, nor even Ven. Brahmali’s matter unless they are substantiated by textual, linguistic and cultural evidence. When it comes to jāti, we should admit it regularly occurs in the rebirth context, so we should really say that is what the Canonical Buddha thought and / or taught whether we can really connect to it or not.
On the other hand, I can’t really see why you can’t observe the different stages of this extended conditioning process encompassing several lives in this very moment. You can observe it even on the conceptual level right down to the consciousness-name-and-form knot, as is mentioned in DN 15. Probably, you cannot adequately describe the workings of sankhara in our language, most certainly you cannot really describe your ignorance. Still, you can observe and analyze the other stages within your conceptual reach and see how they produce each other. Extrapolating the results of your investingation and analysis, you can apply them to your past and future lives to see how the rebirth process works. As Ven. Brahmali mentioned, it is like understanding a law of nature. You see an apple falling down on your head and realize how planets and galaxies are attracted to each other. You observe how your craving is conditioned by your feeling etc., and you can realize how it leads to rebirth. What I strongly agree with Ven. Brahmali on, is that we should not think that the present moment description is the ultimate one, that it is exactly what the Buddha was talking about. It can be a handy heuristic tool for us to better understand the ultimate law, very much like Newton’s apple, but a falling apple is definitely one of the less significant effect of gravity in the Universe.