I am adding some material that supports the idea that a Hindu woman can be independent and even become a monk.
There are examples of women Sannyasis who have attained Moksha in Mahabharata. I will give here 2 such examples:
Pingala said, ‘…What women is there that regards that Supreme Soul as
her dear Lord, even when he comes near? I am now awake. I have been
roused from the sleep of ignorance. I am no longer influenced by
desire. Human lovers, who are really the embodied forms of hell, shall
no longer deceive me by approaching me lustfully. Evil produces good
through the destiny or the acts of a former life. Roused (from the
sleep of ignorance), I have cast off all desires for worldly objects.
I have acquired a complete mastery over my senses. One freed from
desire and hope sleeps in felicity. Freedom from every hope and desire
is felicity. Having driven off desire and hope, Pingala sleeps in
Mahabharata Santi Parva Section CLXXV
In the same Satya Yuga, a woman of the name Sulabha, belonging to the mendicant order, practiced the duties of Yoga and wondered over the whole Earth. In course of her wanderings over the Earth, Sulabha heard from many Dandis of different places that the ruler of Mithila was devoted to the religion of emancipation. ....Sulabha became desirous of a personal interview with Janaka. [ The story is very long. So I will shorten it by saying that Sulabha went to Janaka and entered his mind through her Yoga Power to check if Raja Janaka was truly emancipated. Janaka protested saying that as a mendicant and a woman she should not have entered his body to check whether he has really attained Emancipation. Sulabha gave Raja Janaka a good dressing down. I will quote only a few lines of her retort.]
Sulabha said, 'The contact of one that is emancipated with another
that has been so, or Purusha with Prakriti, can not lead to
intermingling of the kind thou dreadest. Only those that regard the
soul to be identical with the body, and that think the several orders
and modes of life to be really different from one another, are open to
the error of supposing an intermingling to be possible. My body is
different from thine. But my soul is not different from thine. When I
am able to realize this, I have not the slightest doubt that my
understanding is really not staying in thine though I have entered
into thee by Yoga.'
Mahabharata Santi Parva Section CCCXXI
Equal rights enjoyed by women
I am posting a passage from a famous 14th century text which shows the equal rights enjoyed by women in Hinduism.
Renunciation is mentioned in the Taittiriya and other Upanishads:
'Some have attained immortality, not by acts, nor by offspring, nor by
wealth, but by renunciation alone' (Kaivalya Upanishad., 3). Females
also are entitled to this kind of renunciation. The Caturdharike of
the Mokshadharma, by using the word bhikshuki (female mendicant) with
reference to the lady in question, when dealing with the controversy
between Sulabha and janaka, indicated that females may renounce before
marriage, or after the death of their husband and may go about as
religious mendicants, may learn and hear the sastras dealing with
moksha (namely the Upanishad and cognate literature), may meditate
upon the atman in seclusion and assume the emblems of tri-danda, etc.
In the wake of the argument in the Devatadhikarana (the section about
devata) in the fourth chapter of the third book of the Sarirakabhasya,
wherein a discussion has been started relating to the rights of a
widower (to such renunciation), the name of the lady Vacaknavi has
also been mentioned. These references go to strengthen the right
claimed by Maitreyi, the wife of Yajnavalkya, of whom she inquires in
the words: 'Of what use would such (wealth) be to me, if the
possession of such would not render me immortal? Tell me whatever your
worship knows, as to what I should do, in order to be free from
mortality' (Brihadaranyaka Upanishad 4.5.4).
Jivanmuktiviveka of Vidyaranya translated by S. Subrahmanya Sastri and T.R. Srinivasa Ayangar, Chapter 1, Vividisa-samnyasa: The Renunciation of the Seeker