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Anything that inherently exists would not involve change.

Here we have the crux of the issue with the philosophy of the Two Truths. It is predicated on a statement of existence which has no basis in fact. It's like saying that 'unicorns never existed' - well who ever said they did?

In the same way the notion of an unchanging "inherent existence" is put up for demolition. Well who ever said that things exist / don't exist in this manner? We have a speculative ontology here and nothing more than that, despite the rigorous logic of debating philosophers. The concluding 'emptiness' which necessarily follows from this 'analytical process' is merely a logical construction. It has nothing whatsoever to do with Buddha's realisation of Sunyata, which was direct, non-dual and non-discursive.

Okay, it has its uses insomuch as some sects may have insisted on an unchanging atman. It points out the folly of such a position. It also points out the inconsistencies of other Buddhist sects who were likewise wandering into territory Buddha never entered. But it remedies this problem by engaging on the same terms as those erroneous debaters and thus becomes locked into the same paradigm - the assumption of an ultimate basis. It is a handy didactical device designed to win debates, not establish a position on an ultimate state - which was something Buddha assiduously avoided.

Created objects cannot inherently exist since that would involve change.
The emptiness of inherent existence of the mind is called the Buddha nature.
The potential movement of the mind from non-enlightenment towards a state of enlightenment would not be possible if the mind itself existed inherently. The above statements are likewise predicated on the logician's device of "inherent existence".

emptiness is what makes the transformation towards Buddhahood possible. I completely agree here! But emptiness was discovered by Buddha, not devised in debate. It was a significant discovery as it enabled him to go much further than other mendicants of his day in gaining unbinding. Why? Because in all cases where an "ultimate state" is presupposed (even an ultimate state of emptiness), there will be clinging to it at the point where the contemplative assumes that he has encountered it. There are no limits to Buddha's liberation, therefore any such clinging would be detrimental to this unbinding.

Namaste
Kris