How can there be knowledge of unconditioned phenomenon without any knower?

who knows there is an existence of such a phenomenon?

Buddhism says there is not permanent, everlasting, controlling self or a core.

Also, Buddhism says any doctrine about self results in misery

The person who experiences the phenomenon knows it. But having said that there is nothing in the person which can be identified as a permanent core, a part which is everlasting, a part of the person which one can exercise absolute control.

A being in Buddhism is considered dependently arisen from moment to moment. There are billion of such rising and passing which created what is know are a being. A being is not worth being identified as atman as this leads to misery.

how Gautama Buddha knew there is such base/ayantana with those qualities of negations?

This was known through direct experience.

To claim so one has to either experience so or accept the authority of someone, right?

Ultimately this has to be empirically verified.

In order to be motivated to experience, this one has to have some belief in it, to strive towards achieving it.

What is 'it' in 'it has no support'?

This does not depend or supported by another condition. If it was supported by something else if the support break this also falls apart. This does not depend nor is supported by anything else.

It seems some existing substance to me which has been experienced/known thus proving the existence of some known entity.

Since this has been experienced by only a few there is no vocabulary describe it. There is nothing to draw parallels or analogies. What we can say is what we certain this we know are not present. For that is there in the experience there is no language to describe.

Its only emphasized that there is nothing fixed and no abiding self who knows.That doesn't mean that the knowing factor isn't there .There is knowing ,and that knowing has the capacity to reflect even on itself. Thats all no abiding knower ,just knowing exists.

If we ask who is the one experiencing the end of suffering, that is like asking how a photon knows to go in a straight line. Both questions result in a tangle of confusion around the concept of identity and knowing. The conventional meaning of "knows" presupposes identity and without identity view, conventional meaning comes up empty. If we step away from the assumption of an identity or an entity who knows, then we can deal with answerable questions. Let's talk about:

"What is 'it' in 'it has no support?'"

Since Buddhism deals with the end of suffering, the "it" is non-suffering. Suffering is supported by craving. When desire is relinquished, suffering has no support and vanishes.

MN121 illustrates a progression for relinquishing supports and details what happens as the supports are relinquished, one by one:

That’s how emptiness is born in them—genuine, undistorted, and pure.

As you quoted:

Just this is the end of suffering.

This question crops up regularly, and I suspect it's the cause of a lot of worry for those just getting to know the teachings. It was for me at one time. The confusion is caused by the ultimate non-duality of knower and known, experience and experiencer.

Once these distinctions are transcended then we cannot talk about experiencing and knowing in the usual way. Both processes are inherently dualistic.

Hence at profound level of Consciousness and Reality all talk is of Identity, Being and Pristine Awareness. The Upanishads tell us there is no intentional consciousness after death. What there would be is something that probably can never make sense to a person who has spent no time exploring the roots of consciousness.

The crucial point is that Buddhism is not epistemological nihilism. This is a kind of nihilism that is also omniscience. By going beyond the knower-known distinction we would acquire a knowledge of knowledge and experience that is of all knowing and all experiencing, not just of some particular knowledge or experience. Hence the Upanishads also say, 'The voidness of one thing is the voidness of all'. That is, when we understand the nature of one thing we understand the nature of all. In the same way, when we understand the nature of knowing and experiencing we do so for all knowledge and all experience.

Bernardo Kastrup has a highly relevant poem on youtube called 'Legacy of a Truth-Seeker'. This is not really an explanation of anything but it covers a lot of the ground. https://www.youtube.com/watch?v=ZgiwVYZM5A8

I feel your opening statement is not quite right. It is the Perennial view that what is fundamental and original is knowing. This is what distinguishes or characterises consciousness. But this would not be a knowing that requires a division into knower and known. It is not what we usually call knowing but Being, the simple fact of 'I Am' that Descartes spotted was not relative and could not be doubted. This 'knowledge by identity' would be the basis for all knowing for the knower would be the identity, the knowledge the knower. There would be nothing to know except the knower.

Theists capture something like this idea when they speak of God spending eternity basking in His own glory or contemplating on His own nature. The connection to Buddhism can be made by way of the idea of 'Godhead', not a God but a state or 'hypostase' of Consciousness and Reality.

This is about 'pure knowing'. Rupert Spira gives a talk on the topic here. https://www.youtube.com/watch?v=GhCcYU_vzIM

He doesn't rush so you may need to make a brew and settle in.

In the end all you can do is test the teachings in your own practice, but it would be important to avoid the idea they are pessimistic.

The sutta quoted says Nibbana is an "ayatana" or "base". Therefore, it is a sense object.

The "knower" of Nibbana is mind-consciousness rather than a "self". This "knower" is a "nama-dhamma", "mental element" or "conditioned element".

Nibbana itself is the "unconditioned element" ("asankhata dhatu" - MN 115) therefore the Nibbana Element is "unsupported" by any condition.

"Consciousness" or the "knower" only "knows" Nibbana. The "knower" does not "support" Nibbana.

In short, the enlightened conditioned consciousness, while living, knows the unconditioned Nibbana.

When the conditioned consciousness ceases to exist, the unconditioned Nibbana, as an element (rather than as a known sense object) continues to exist.

The element of Nibbana is eternal & permanent. Knowing consciousness is impermanent & conditioned.