I have read very little of Dogen.
Then your remark...
As I have already mentioned before, in the zen tradition all these "core teachings" concerning the Four Noble Truths, the Eightfold path, and a ton of Buddhist theory and practice is all utilised with little or no reference to rebirth - it is just not relevant to this tradition. A zen teacher would ask you to focus on the here and now, rather than debate on if rebirth is true or not, it just wouldn't be a factor at all in defining one's Buddhist practice.
...is based on the statements made by who? Modern representatives of either Soto or Rinzai?
[i]I don't understand that if there is no atman (permanent self) beside this impermanent body and mind, what is chanting, "I take refuge in Buddha." after the death of this body and mind? Anyway, if this is a contradiction, Buddhism itself has had this contradiction from the very beginning until today. Many Buddhist philosophers have tried to clarify this point and no one has been completely successful.
He seems to have no understanding of how only whereby lacking atman can any being at all exist due to the process of causality or dependent origination. Furthermore, he seems to assume the mind, being impermanent, is tied to the demise of the body.
I am not going to try to create a new theory to explain this contradiction. I don't believe in rebirth and yet, I don't negate it. There is no basis to believe or negate it. What I can say for sure is, "I don't know." The important thing for me is to practice in this lifetime as the Buddha instructed in the Dhammapada, "To refrain from anything bad and practice everything good. Purify your mind.
Anyone from any religion can purify their minds, but without right view it is all futile. One might achieve the highest peak of existence in the arupa-loka, but the ultimate result is falling back down onto the rougher side of the wheel of life.
Also it is curious that he is quoting a Pali scripture and saying following the Buddha's instructions is important to him, yet he is denying a fundamental teaching of the Buddha. I suppose he is picking and choosing which teachings he likes? If anyone is guilty of contradiction, it is him. To say the Buddha's instructions are important and then be agnostic about rebirth and denying its importance is complete foolishness.
One of the points that I would make from all this, is that many aspects of Buddhist teaching and practice are not so easy to pin down as we might think.
If one properly studies, reflects and practises both meditation and morality, then the whole process of being a mere dweller in samsara to Buddhahood (or Arhatship in the case of Sravakayana) can be neatly mapped out and this has been done in numerous traditions.
Knowing the work that I have to do on myself, I find it incredibly hard to know how I could really judge others so clearly, and precisely, when in reality, my own view is partly obscured! And, I am accepting and rejecting different ideas as I go along, trying to make my way through the forest, just like the Rev Okumura, and countless others.
Such vague and sympathetic remarks are a new trend in Buddhism. If you read actual scriptures and texts you'll almost always find the authors (of any period or culture) saying without reservation their thoughts and making clear unsympathetic judgements about others and more often than not the judgements are very critical and harsh. These are so old and usually in dead languages that few would find them personally offensive, but if you raise the same arguments against people now you'll be called insensitive, judgemental and arrogant.
In any case, dwelling on how you're wandering through a forest and how remarkable such a spiritual path of uncertainty is your freedom, but don't expect me to be sympathetic towards such sentimental nonsense. The Buddhadharma if properly understood is an almost mechanical process that functions as a therapy where the problem of suffering is addressed and the cure is prescribed through various means which are explained in detail without hesitation.
Abiding in a "forest" and feeling wonder at your own uncertainty while praising others own doubtful attitudes is hardly a reasonable and rational approach towards the disease of samsara. What you describe is more like a visit to judgement-free self-help session than an actual methodology to end suffering.