Why Hindu Brahmins were not allowed by other castes,which includes Kayasthas to read BHAGVAD GITA and were (discouraged by Hindu Bramhan Gurus also}during a distinct period of history? Is there any specific reason?
There are no such restrictions whatsoever. Krishna DvaipAyana VyAsa (also known as Veda VyAsa) was a Maharshi & son of a Brahmana named ParAshara. Sage VyAsa himself had passed on Gita (as part of Mahabharata) to his [supposed] Brahmana disciples.
Moreover, with the context of Moksha and Dharma, Shri Krishna declares that the study of Gita (learning or hearing) with faith, is virtuous:
BG 18.70 - And those who will study this sacred conversation between us two, which is conducive to virtue, by him I shall be adored through the Sacrifice in the form of Knowledge. This is My judgement..
BG 18.71 - Any man who, being reverential and free from cavilling, might even hear (this), he too, becoming free, shall attain the blessed worlds of those who perform virtuous deeds.