Why can any player use a wand, but magical staves have class limitations? Is there a mechanical reason for it (like balance), or is it purely for flavor / game design?
It is for historical reasons based on the intent of E. Gary Gygax when he designed Advanced Dungeon & Dragons. There are a couple of staff items not found in classic D&D and it looks like the 5e author followed Gygax's lead.
In OD&D the Staff of Healing and the Snake Staff were both noted as a clerical. When Gygax created AD&D he expanded this to restrict the staves to certain classes.
Likely he did this for flavor reasons. For example, the Staff of Healing and the Snake Staff evoke some of the Bible stories and certainly De Mille's Ten Commandments with Heston had an impact on the popular image of staves that can heal and turn into snakes. The Staff of Power and the Staff of Wizardry evoke various traditional tropes held about wizards.