Is there a place for someone who says "Whether or not he is someone's reincarnation doesn't matter, this Rinpoche is wise and I will learn from him what I can" to practice sutrayana-only, or is belief in reincarnation/tulkus and Vajrayana practice necessary for this path?
Buddhism is not about being in the club of right beliefs. It is fundamentally about offering a path of benefit and happiness to all who wish for it, without qualification or exception.
Part of this part is shaped by right view (right view here is defined as 'right' insofar as it leads to a deeper happiness), but views on rebirth is only a part of this matrix of views. You can still gain tremendous benefit from Buddhist practices without having embraced all aspects of right view (very few practitioners manage such a feat), so if you are one who wants to pursue this there is no reason you shouldn't.
Finally, Buddhism does not ask for blind faith. Belief is a part of Buddhism, but it is not unquestioning. Belief in Buddhism is held to be most skilful when it is a consequence of contemplation, investigation and support from personal experience. I think if you keep an open mind willing to consider how rebirth could
be true, you should feel no compulsion to adopt belief in rebirth or bad about not finding the arguments for it compelling enough to accept.
Generally, it is considered that the more you practise and and come to understand the Buddha's teachings through personal experience, the more logical the other parts of the teaching also become, even if you do not yet have personal experience of it. It is this kind of belief through understanding that Buddhism encourages, even as it maintains that belief is only a pragmatic and temporary placeholder for the truth of personal experience and not to be accepted as actual truth.