Madhyamika, nature of mind, Advaita, etc
Modern and traditional Advaita alike don't really teach about emptiness as the interdependence and lack of essence or substance of self and things... it is not just modern Advaita which takes ultimate reality to be some substantial "foundational background" or "ground of being" or "source and substance of all that is".
Brahman, their ultimate reality, is traditionally known as Sat Chit Ananda - Existence, Consciousness, Bliss. It is seen as substantially existing, not dependently originated, the ultimate true Self (Atman-Brahman) of all beings and as some irreducible Truth that is the summum bonum of spiritual life.
Buddhists and 'emptiness teachings' would disagree that 'Consciousness' is something 'irreducible'.
Right from traditional Advaita and the Vedas and Upanishads, we find a living tradition that treats the world as illusory and Brahman as the ultimate reality, i.e. it attributes 'forms' to be 'illusory Lower truth' as a mere illusory appearance of Brahman the 'higher truth'... and even in the final position that 'the world is Brahman' it actually means 'the world is but an appearance inseparable from the ultimate higher truth of Brahman, like printed words is inseparable from the paper underlying it, and apart from the ultimate reality of Brahman or Pure Consciousness there is no objective reality anywhere'.
This is not what Emptiness teachings are talking about.
Emptiness does not 'reduce' the world or objectivity into a substantial source and substance but liberates it, it frees us from erroneous imputation and grasping of I, me, mine, and objects.
The only living contemplative tradition that talks about emptiness in a mere deconstructive (not reductionist) way, and in terms of interdependent origination, would be Buddha and the Buddhist tradition such as Madhyamika.
It may be argued that some philosophers since have adapted or taught doctrines that may have similar elements to the Buddhist teachings, however I think Buddhism so far offers the most indepth deconstruction and most of all a practical method of contemplation aimed at directly seeing the truth of it in experience.
And the Buddha would certainly be the first (2,500 years ago) to teach a non-substantialist doctrine of emptiness.
In any case, it is quite different from other spiritual and contemplative traditions we see out there, including Advaita Vedanta.
And since Advaita treats Brahman, pure consciousness, as ultimate reality, Greg Goode calls Advaita not an "Emptiness teachings" but an "Awareness teachings".
I think that is a more appropriate description of Advaita.
The Buddhist understanding of emptiness, middle way, and two truths, though sounding similar is nevertheless vastly different from the usage by Shankara and Advaita Vedanta - whether traditional or neo. I.e. Buddhism's two truths does not mean that phenomena are the lower truth in contrast to the highest truth of ultimate reality or Brahman or Awareness.
Instead, there is no 'higher' or 'lower' - the two truths refer to the ultimate truth and the conventional, imputed truth - that means all our labels like 'self', 'table', 'weather' are mere parlance and imputation - a conventionally labeled designation commonly agreed upon... but the ultimate truth (not exactly 'reality') of things is that when investigated, there is no real substance or true, independent existence graspable behind those conventions or appearance - all of which are interdependently originated.
That lack of true essence (svabhava) is the ultimate truth of all phenomena, and emptiness is 'designated' dependent on phenomena's lack of essence/dependent origination, apart from phenomena no 'emptiness' can be spoken of.
Therefore there is no way emptiness can be called a 'higher truth' and phenomena a 'lower truth', which is the way it is taught in the traditional Advaita Vedanta teachings. Instead, the Heart Sutra puts it succinctly:
Form is Emptiness,
Emptiness is Form.
Everything is flat without hierarchy.
Lastly... and I think this is a most important point, I think emptiness and luminosity need not be seen as two different approaches. In fact we should see the inseparability of luminosity and emptiness.
That is we have to see the luminous essence or quality of mind/experience - that everything is vivid luminosity, vivid cognizance, vividly self-aware.
In fact the peak of this luminosity is experienced when all divisions of subject/object, perceiver/perceived dichotomy is completely seen through... through seeing through there is a natural non-dual opening to vivid sensation in every moment and experience.
But as vivid and luminous and aware as mind/experience is, its empty, unlocatable, ungraspable, coreless nature of everything - 'selves' and 'things', all phenomena, must also be penetrated for true liberation... for luminosity is blissful, but emptiness is liberating. Which is why it is more impactful, in the sense that it releases all attachments and graspings due to freedom from establishments - non-dual luminosity on the other hand can even lead to greater attachment to some ultimate non-dual luminous state.
But non-dual luminosity and emptiness is inseparable, and this inseparability (of appearance and emptiness, bliss and emptiness, luminosity and emptiness, etc) is given many names in the Tibetan traditions such as 'the nature of mind', 'Buddha-nature', and so on.
Madhyamika emphasizes the aspect of understanding the truth of emptiness through logical reasonings... but Tibetan Buddhism utilizes Madhyamika as only part of their training program. Madhyamika is important for them so that they have the 'Right View' that does not end up in a substantialist view such as Advaita and at the same time, they engage in meditative training such as Mahamudra, Dzogchen, etc which focuses on training on recognising the nature of mind (as the inseparability of luminosity and emptiness, rather than just remaining at the level of conceptual understanding of emptiness).
While Mahamudra is most often attributed to the Kagyu tradition and Dzogchen with the Nyingma tradition, the Sakya and Gelug traditions also engage in a myriad of tantric methods to realize the Clear Light essence of mind and the inseparability of luminosity and emptiness through those practices such as those involving the two stages of tantric practices (creation and completion stage), the result of which is none other than Mahamudra.
The means may differ slightly but the goal is always the same.
For them, Madhyamika is mostly like a foundation for establishing the view of emptiness albeit on an inferential/conceptual level, and then they engage in tantric methods to quickly realize the inseparability in direct experience.
