I'm a great fan of Swami Sivananda, whose work you quoted. He was a very old-school type Guru, and remarks such as those are made in the context of the predominant cultural environment (patriarchal Hindu India, in this case). Actually, Hinduism in its original form as Sanatana Dharma accorded women equal power (and in some lineages greater power than men). One of the highest esoteric practices is worship of the Divine Mother as creatrix of the universe, second to none
In India, the southern state of Kerala, which gave birth to some of its greatest saints, is a matriarchal society to this day. Unfortunately, chauvinistic males have dominated religion, and excluded women in a display of power (stemming from insecurity and ego!).
The new spirituality which is being ushered into world consciousness shall be shaped by the feminine aspect of the Divine: compassion, sweetness, humility, tolerance, and grace under pressure.