It goes too far. The reincarnate lamas are recognized as being the same person as someone who passed away before their (re-)birth in the current lifetime. So clearly, a "self" was reborn. The alaya vijnana is said to be such very subtle consciousness, it's mainly only karmic seeds; the consciousness is believed to be too subtle to bring with it past-life memories, a personality, claims to ownership of accoutrements from the previous lifetime/s, all of which smack of a "self". On the other hand, the Buddha is said to have gained total recall of his past lives, so that doesn't square with the descriptions/explanations of alaya vijnana I've heard/read. Go figure.
None of that indicates anything about anything but a conventional self, it indicates habits and inclinations which persist from one lifetime to another. The idea of post mortem rebirth and having/retaining similar habits indicating a "self" is a misunderstanding IMO, there is actually a good description of the accumulation of habit-energies in the Lanka. Basic notion is, you will be reborn according your actions of body, speech, and mind, so there is continuity. It is not static, but of course there is repetition of similar patterns - that's Samsara after all.
This is why in Buddhist circles a distinction is drawn between "rebirth" vs. "reincarnation". "Reincarnation" is generally considered to be a Hindu concept or phenomenon, highly controversial in a Buddhist context, precisely because it implies that a "soul" or identity is carried on. Vajrayana is the only Mahayana tradition that has reincarnation. According to historians of Buddhism, it developed at a time when Indian Mahayana experienced strong influence from Hindu traditions, like tantrism among others, and when Buddhism was losing followers to Hindu traditions, so it adapted in order to compete. (See Ronald Davidson, among others.) So it's possible that reincarnation came into Buddhism at that time, and was reinforced when introduced in Tibet.
I think that's a bit off.. the idea of having similar predispositions from life to life is basic to Buddha dharma, and any understanding of Karma, it doesn't indicate believing in reincarnation over rebirth at all, though there are some that claim Yogacara is a kind of proto-atman view, I don't agree. basically you are saying "because people are (re)born with similar attributes, this must indicate a self..that's off the mark, the similarities are explained by Karmic habituation..not by any kind of self. The same goes for Tulkus, etc. in Vajrayana. No one is saying there is no such thing as conventional selves, we all have those, and we wouldn't be "beings" without having them.
But you've intrigued me with your statement that in Bon, the "soul" is not like a self. Could you elaborate on that?
I have only a bit of familiarity with Bon, but suffice to say it is not the unchanging thing we are talking about as a "self', I believe it's more something that'd be termed a "life energy" than some kind of immutable soul.
As to Tantra, there is one level of Tantra that is intentionally similar in outer practice to Hinduism, beyond that they diverge philosophically, and Buddhist Tantra is in fact, Buddhist ...whatever academics speculation of the month on Tantra happens to be. Anyway, AFAIK there is no real agreement on whether Buddhist or Hindu Tantra came first, and I don't think anyone has the definitive answer.