If Buddha really combated and denied all Vedantic conceptions of the Self, then it can be no longer true that Buddha refrained from all metaphysical speculations or distinct pronouncements as to the nature of the ultimate Reality. The view you take of his conception of Nirvana seems to concur with the Mahayanist interpretation and its conception of the Permanent, dhruvam, which could be objected to as a later development like the opposite Nihilistic conception of the Shunyam.
What Buddha very certainly taught was that the world is not-Self and that the individual has no true existence since what does exist in the world is a stream of impermanent consciousness from moment to moment and the individual person is fictitiously constituted by a bundle of samskaras and can be dissolved by dissolving the bundle. This is in conformity with the Vedantic Monistic view that there is no true separate individual. As to the other Vedantic view of the one Self, impersonal and universal and transcendent, it does not seem that Buddha made any distinct and unmistakable pronouncement on abstract and metaphysical questions; but if the world or all in the world is not-Self, anatman, there can be no more room for a universal Self, only at most for a transcendent Real Being.
His conception of Nirvana was of something transcendent of the universe, but he did not define what it was because he was not concerned with any abstract metaphysical speculations about the Reality; he must have thought them unnecessary and irrelevant and any indulgence in them likely to divert from the true object. His explanation of things was psychological and not metaphysical and his methods were all psychological,—the breaking up of the false associations of consciousness which cause the continuance of desire and suffering, so getting rid of the stream of birth and death in a purely phenomenal (not unreal) world; the method of life by which this liberation could be effected was also a psychological method, the eightfold path developing right understanding and right action. His object was pragmatic and severely practical and so were his methods; metaphysical speculations would only draw the mind away from the one thing needful.
As to Buddha’s attitude towards life, I do not quite see how “service to mankind” or any ideal of improvement of the world-existence can have been part of his aim, since to pass out of life into a transcendence was his object. His eightfold path was the means towards that end and not an aim in itself or indeed in any way an aim. Obviously, if right understanding and right action become the common rule of life, there would be a great improvement in the world, but for Buddha’s purpose that could be an incidental result and not at all part of his central object. You say, “Buddha himself urged the necessity to serve mankind; his ideal was to achieve a consciousness of inner eternity and then be a source of radiant influence and action.” But where and when did Buddha say these things, use these terms or express these ideas? “The service of mankind” sounds like a very modern and European conception; it reminds me of some European interpretations of the Gita as merely teaching the disinterested performance of duty or the pronouncement that the whole idea of the Gita is service. The exclusive stress or overstress on mankind or humanity is also European. Mahayanist Buddhism laid stress on compassion, fellow-feeling with all, vasudhaiva kutumbakam, just as the Gita speaks of the feeling of oneness with all beings and preoccupation with the good of all beings, sarvabhuta hite ratah, but this does not mean humanity only, but all beings and vasudha means all earth-life. Are there any sayings of Buddha which would justify the statement that the object or one object in attaining to Nirvana was to become a source of radiant influence and action? The consciousness of inner eternity may have that result, but can we really say that that was Buddha’s ideal, the object which he held in view or for which he came?
There is no reason why the passage about Buddhism should be omitted. It gives one side of the Buddhistic teaching which is not much known or is usually ignored, for that teaching is by most rendered as Nirvana (Sunyavada) and a spiritual humanitarianism. The difficulty is that it is these sides that have been stressed especially in the modern interpretations of Buddhism and any strictures I may have passed were in view of these interpretations and that one-sided stress. I am aware of course of opposite tendencies in the Mahayana and the Japanese cult of Amitabha Buddha which is a cult of bhakti. It is now being said even of Shankara that there was another side of his doctrine—but his followers have made him stand solely for the Great Illusion, the inferiority of bhakti, the uselessness of Karma—jagan mithya.