Have you seen evidence that Gautama Buddha visited Shambhala?

There are too many later added mysticism attributed to the Buddha, when clearly anyone with logic can see that they are religious dilutions or additions, perversions of the Buddhas teaching.

Tibetan Buddhism as a whole is rife with these type of perversions. From basic doctrine additions attributed to the Buddha, like nagarajunas philosophy then written as the heart sutra and told to have been written by the Buddha, kept secret by naga spirits until the "right time" when people were ready for it and found by nagarajuna in a lake... to the entire pantheons of deities supposed to be Buddhas in different worlds, or dimensions, or realms. Heck there is even attributes to Buddhas teachings which asanga added (creating more mental formations of consciousnesses) and said to have been taught to asanga by matreya in the brahma realms.....Yes it really is this perverted.

More recently there is a general schism created in Tibetan Buddhism as a whole via belief in a fictional deity, dorje shugden. Let alone Tibetan Buddhism itself is unable to unify the Buddhas teachings and have 4 major (and many minor) sects within the country that vastly differ in what IS and what IS NOT the Buddhas teachings.

The point is, over 2500+ years since the Buddhas death, there have been countless people claiming they are a Buddha, or have new teachings from the, or a new Buddha, or written teachings as if it was the Buddha talking, or changed the words of the Buddha in sutras... Essentially it is chinese whispers.

A lot of this added mystical elements originate from hinduism, which in a lot of cases have converged, where just simple name changes (like adi-Buddha and vishnu) of hindu deities have "allowed" these deities to then be attributed to the Buddha-dhama.

It would be wise for you yourself to be able to discriminate, as per the kalama sutta as to what is really needed and what is to be practised, and to rid yourself of unnecessary mystical elements that bring no benefit.

Shambhala like Mount Kailash, is best understood as a spiritual or metaphysical location, which at best only correlates weakly with a real physical place. We might compare it with Atlantis, or Camelot, as a place with pretty sketchy and minimal 'original' mentions, which went on to take up an outsize role in historical and mythological discussion. I think we have to understand that kind of dynamic, as how an idea can speak to a wish we have, to something many people want to be in the world or to show something about the world from it being part of history.

The core prophecy about Shambhala, derived from the Kalachakra or Wheel of Time sutra, has to be unserstood in the context of the decline of Buddhism in the Indian subcontinent, happening around the time modern scholars considered the sutra to have been written (1025-1040 CE). Especially, the decline of Nalanda Mahavihara and the Great Library there which formed the centre of what became Mahayana Buddhism and the transmission of Buddhism to East Asia. Shambhala symbolised a place of security in retreat for Buddhism, as Islam was spreading, and especially persecuting the way of life and funding for Buddhist monastics.

Consider the Four Buddhist Persecutions in China, which caused the monastics that survived to retreat to the high mountains, to such an extent the Chinese word for monastery is 'shan', meaning mountain. The Zen Golden Age, followed the Great Huicheng Persecution, which can be understood as a kind of stripping away of the wealth and social approval for Buddhist practice, requiring a focus on the very core of the practice, worth continuing to do even at risk of life and liberty. So a monastery becomes a mountain even if it's in a city, a place of retreat and purification.

Mahayana and Vajryana Buddhism have literally had this journey, of retreat into the Chinese and Tibetan mountains, and a kind of recentring and refocusing of purpose, followed by a spread back out into the world after a position of weakness and near-extinction, resulting from their new clarity of teaching and purpose.

The Shambhala tradition and the Maitreya tradition, implicitly says 'real' Buddhism will (almost) cease, and the kind of practice that leads to awakening or the full teaching however that is framed, will need delivering again, or renewing from the place where it has survived. Humans change, society changes, and the challenges we face change. So I would understand Shambhala as a metaphor for the renewal of Buddhism, the retreat from rich trappings and social approval, into conditions of simplicity and confrontation with the purpose our practice serves, from which we can renew the teachings and restore the vigour of our practice and the directness with which it speaks to new times.

The real Shambhala, is the true teaching seperated from rewards and approval which over time can obscure it. It is a place that's to be found within all practicioners. And of course the Buddha has been there, just as the sutras describe him going to Tushita Heaven, and delivering the (Pali) Brahmanjali Sutta teachings to Maha Brahma. It is a description of spiritual not literal geography.

No, and you're not likely to find any. The concept of a Shambhala in Buddhism comes from the Kalachakra tradition, which originated sometime in the 10th or 11th century. You haven't found any evidence because the concept didn't exist while the historical Buddha was alive. If Shambhala is taken to mean Sambhal (UP), as per some of the early Puranas, then the historical Buddha was already in that region (Kushinagar), though I'm unable to find when the town of Sambhal was actually founded.