Are Vedanta and other systems derived purely through logic or do they have an experiential basis too?
You have asked 2 questions.
The first question is what will a person realize without knowledge of Hindu metaphysics.
This is an excellent question. To answer this we have to know that there are essentially three different types of meditation: meditation on eternal subject, meditation on eternal object and finally meditation on neither.
The source of eternal subject meditation is an Upanishadic shloka like that given below:
"The eye does not go thither, nor speech, nor the mind. We do not know It; we do not understand how anyone can teach It. It is different from the known; It is above the unknown. Thus we have heard from the preceptors of old who taught It to us."
(Kena Upanishad I.3-4)
The above shloka is talking about the Atman that is an eternal subject. The two essential methods to inquire about the Atman are (1) the "neti neti" method and (2) "who am I?" method. In the 'neti neti' method all objects are negated until one reaches the eternal subject the Atman. In the 'who am I?' method all attributes of oneself is negated until the Atman is realized or Brahma Jnana is attained.
The source of eternal object meditation is an Upanishadic shloka like that given below:
I know the Supreme Person of sun-like color (lustre) beyond the darkness. Only by knowing Him does one pass over death. There is no other path for going there.
Svetasvatara Upanishad III.8
This shloka is talking about God as an eternal object. The usual method to attain God as an object is to do rupa dhyana or meditation on a form.
The third type of meditation is the Buddhist one which avoids both meditation on eternal subject and eternal object. If both the subject and object are avoided then you are left with Sunya or nothing. Buddhists claim that this meditation will lead to nirvana.
I have now discussed the theoretical point of view. In reality from the practical point of view a Guru is necessary for any spiritual practice and a spiritual aspirant will pick up knowledge of the metaphysics from the Guru. It would seem that the spiritual aspirant would know some metaphysics even if the metaphysics is not of the Hindu variety and will have realization suitable for that metaphysics.
The second question is whether Vedanta and other systems have an experiential basis.
They have an experiential basis and shastras say that the personal experience is far more important than intellectual knowledge of shastra or scripture.
They study the Vedas and discuss. But they do not realize the Ultimate Reality just as a spoon does not know the taste of food.
The head carries the flowers, the nose knows the scent. The people study the Vedas. But, very few persons understand the same.
Not knowing the Reality of the self, a fool is infatuated by the shastras. When the goat stands in the shed, the shepherd seeks for it in the well in vain.
The knowledge of the shastras is not competent to destroy the infatuation accruing from worldly affairs.
Having studied the Vedas and realized their essence the wise man should leave all the shastras just as one desiring corn leaves the husk.
Just as one satiated with nectar has no use of food, no one who is in search of Reality has anything to do with the shastras.
One cannot obtain release by reading the Vedas or the shastras. Release comes from experience, not otherwise, O son of Vinata.
[Garuda Purana, Dharma Khanda, Chapter XLIX]
Logic / Reason / Science not sufficient to realize God
In many of his speeches, Swami Vivekananda states that logic is insufficient in achieving Moksha or realizing God. His premise is that logic is made assuming set if facts. This set is very limited given that only so much can be known by observing external world. Although from time to time new facts emerge and enhance our understanding, yet it is insufficient.
Therefore, Swami Vivekananda recommends to look inward.
The Prana is the vital force in every being. Thought is the finest and highest action of Prana. Thought, again, as we see, is not all. There is also what we call instinct or unconscious thought, the lowest plane of action. If a mosquito stings us, our hand will strike it automatically, instinctively. This is one expression of thought. All reflex actions of the body belong to this plane of thought. There is again the other plane of thought, the conscious. I reason, I judge, I think, I see the pros and cons of certain things, yet that is not all. We know that reason is limited. Reason can go only to a certain extent, beyond that it cannot reach. The circle within which it runs is very very limited indeed. Yet at the same time, we find facts rush into this circle. Like the coming of comets certain things come into this circle; it is certain they come from outside the limit, although our reason cannot go beyond. The causes of the phenomena intruding themselves in this small limit are outside of this limit. The mind can exist on a still higher plane, the superconscious. When the mind has attained to that state, which is called Samâdhi — perfect concentration, superconsciousness — it goes beyond the limits of reason, and comes face to face with facts which no instinct or reason can ever know. All manipulations of the subtle forces of the body, the different manifestations of Prana, if trained, give a push to the mind, help it to go up higher, and become superconscious, from where it acts.
Complete Works 1.3.3 : Prana
Need of a Guru
Swami Vivekananda says that although God resides in each of us, an external help to realize it quicker is not a bad idea. In fact, it may be needed in vast majority of cases.
Every soul is destined to be perfect, and every being, in the end, will attain the state of perfection. Whatever we are now is the result of our acts and thoughts in the past; and whatever we shall be in the future will be the result of what we think end do now. But this, the shaping of our own destinies, does not preclude our receiving help from outside; nay, in the vast majority of cases such help is absolutely necessary. When it comes, the higher powers and possibilities of the soul are quickened, spiritual life is awakened, growth is animated, and man becomes holy and perfect in the end.
Complete Works 3.2.4 : The need of Guru
He says that it is temporary, even dangerous sometimes to explore Moksha on our own. Interest in dharma or God is usually triggered by some event, usually an emotional one, which may or may not be true realization. If someone is really hurt on losing someone dear, we look towards God with more confidence. However, as the grief passes, the commitment towards God also fades. Hence these are temporary.
There are, however, certain great dangers in the way. There is, for instance, the danger to the receiving soul of its mistaking momentary emotions for real religious yearning. We may study that in ourselves. Many a time in our lives, somebody dies whom we loved; we receive a blow; we feel that the world is slipping between our fingers, that we want something surer and higher, and that we must become religious. In a few days that wave of feeling has passed away, and we are left stranded just where we were before. We are all of us often mistaking such impulses for real thirst after religion; but as long as these momentary emotions are thus mistaken, that continuous, real craving of the soul for religion will not come, and we shall not find the true transmitter of spirituality into our nature. So whenever we are tempted to complain of our search after the truth that we desire so much, proving vain, instead of so complaining, our first duty ought to be to look into our own souls and find whether the craving in the heart is real. Then in the vast majority of cases it would be discovered that we were not fit for receiving the truth, that there was no real thirst for spirituality.
Complete Works 3.2.4 : The need of Guru
Swami Vivekananda asserts that only a ripe soul can walk on path of realization. Until then, it is just a playful exercise.
...All others are only playing with spirituality. They have just a little curiosity awakened, just a little intellectual aspiration kindled in them, but are merely standing on the outward fringe of the horizon of religion. There is no doubt some value even in that, as it may in course of time result in the awakening of a real thirst for religion; and it is a mysterious law of nature that as soon as the field is ready, the seed must and does come; as soon as the soul earnestly desires to have religion, the transmitter of the religious force must and does appear to help that soul. When the power that attracts the light of religion in the receiving soul is full and strong, the power which answers to that attraction and sends in light does come as a matter of course.
Complete Works 3.2.4 : The need of Guru