Do psychic abilities appear when one becomes an arahant, or is the mind simply as it was before but free of the Taints?
Psychic abilities are a result of Kamma.
An example of that can be found in the case of Anuruddha. In the Therāgāthā – the stories of the early arahant monks – it says that Anuruddha lit an oil lamp for one of the previous Buddhas, Sumedha, while the latter would meditate beneath a tree. As a kammic result, in his final rebirth, Anuruddha was praised by the Buddha Gotama, as being foremost in developing the Divine Eye. He was able to see beyond the physical eye, viewing beings in a thousand world-systems.
One may be able to develop such psychic abilities, whether one has entered the Stream or not, and whether one has completed the Noble Eightfold Path or not.
It is a matter of practice and unlocking such abilities through the development of mind, using the jhanas.
However, final knowledge of the Destruction of Craving / Ignorance / the Fetters and Taints, supersedes all other abilities.
An arahant could develop them, but would not have an inclination to develop these abilities.