Classes primary set a role to play. They offer special abilities and skills which distinguish them from other classes.
The only difference is the starting equipment.
Where is the role playing part if a barbarian is able to become a mighty spell caster; or a sorcerer a skillful sword fighter?
Solution
Remove the class as an attribute of the character and replace it by something like a title. “You are what you do” will be the new motto. Classes as a concept will still exists, but won’t be a fixed attribute of the character. Instead of that the character will join a class guild (see guilds).