> None of the verse you cite says that that indestructible thing is the consciousness

The Self IS consciousness. It is sat - real, that which always is, never is not. cit - It is conscious. ananda - blissful.

Consciousness isn't what you think it is, maybe you're referring to the western definition of it which is as unreliable as anything. It's not your memory, it's not your ego, it's not your mind. These things are a part of prakṛti, or material nature and they're born of the three modes of material nature as explained in the Gita. This metaphysics is a part of the sāṁkhya philosophy.

This is subtle knowledge, it's not easy to understand these abstract ideas without a good teacher to explain it to you or some good amount of self-reflection and study. I'd highly suggest you to read Nyāya sūtra and the Bhagavad Gita (preferably with a teacher who practices these things) for a logical understanding.

> For everything that has an opposite, there would be a beginning and an end

Only the things in the material nature are composed of duality. Gita's whole point is about you, the Self, who is different from material nature, is only covered by it.

> consciousness vs unconsciousness

You're confusing the mind with consciousness. Mind is a part of material nature. There's no such thing as unconsciousness in the Gita philosophy. There are material things and there's the eternal Self, the consciousness.

> Even if there is something eternal, then that also has an ending

That's a philosophically invalid argument. Eternal by definition means that which always is. The concept of cause and effect does not apply to 'eternal' things, it is beyond time and beyond our understanding.

> because in my experience that doesn't exist now, hence it has ended.

In your experience, the future doesn't exist either, has it ended? This argument of yours is called 'hasty generalization fallacy'.

How can the Self not exist? It's literally You. The Self exists right now, it will always exist. If you think the observer dies with the material body, how did the observer come to be in the first place? An observation exists only in relation to an observer, not independently of it.

> It may come back, but doesn't mean it is eternal.

Eternal means existing forever, without beginning or end. The Self can go through different bodies, different experiences, but it 'never not exists'.

> Something that comes and goes means that is not eternal: it is changing too.

That is material nature. Intelligence, ego - false identity, the mind, sense organs, organs of actions, subtle senses and the material body and the elements that make up the material nature, all have a beginning and an end. Consciousness is even subtler than all of these things, it is not a part of the material nature, it's only covered by it.

I'd highly suggest you to study these things in detail, from a good teacher, with logic. Mental speculation is pointless, philosophical speculation provides a better understanding.

Edwin Bryant has a fantastic 8 part series on Yoga Sūtras where he goes over each thing in detail. He also has a series on nyāya sūtras which explains everything using logic. You can refer to that: https://sites.rutgers.edu/edwin-bryant/yoga-sutras/