If I were designing the next edition, I would:
1.) Combine the Barbarian and Fighter. There is not enough distinction between the core of the two classes. Adding the barbarian unarmored defense and rage at 3rd level would also limit the 'dip it to win it' mentality.
2.) Add Artificer and Psion to the core classes. These each fill roles that are not addressed by existing classes (when you execute them well). My versions would be more aligned to the archetypes than we've seen in recent editions. A Psion would not, for example, use spells like a cleric, druid, bard or arcane spellcaster. Nor would Artificers - who would use steampunk science to build contraptions that function like magic items, but do not rely upon magic.
3.) Combine the Ranger and Rogue. I believe we lose a little bit of the core concept of these classes when we try to differentiate them - so don't. Make a single stealthy class that deals extra damage through hitting the sweet spot - but use subclasses to take them the route of the natural spellcaster, the thief, or the beastmaster. I would also roll the concept of the 4E Avenger in here as well as a sublass option.
4.) Combine the Cleric and Paladin. There are too many times when we artifically differentiate a concept. There is no need to do so here - a Holy Warrior is a Holy Warrior.
5.) Combine Sorcerer and Wizard - but primarily for administrative purposes. I'd have these spellcasters have the ability to harness innate magics if they gain access to them (through heritage or corruption of their bodies), but also to use their intellect to master magics. Regardless of which way they accessed magics it would pull from a single source of spell power available to them. Some would focus only on study and mastery while others would go for the simplicity of utilizing innate magics. Psion is more distinct - to me - than Sorcerer and Wizard are.
6.) I'd have the Warlock absorb some of the concepts of the Blood Hunter.
That would give me 10 classes: Artificer, Bard, Cleric/Paladin, Druid, Fighter/Barbarian, Mage/Sorcerer/Wizard, Monk/Psychic Warrior, Psion, Rogue/Ranger, Warlock/Blood Hunter.
I'd also redesign spellcasting so that we start to see higher level spellcasters be able to cast lower level spells an unlimited number of times - turning 1st, 2nd and eventually 3rd level spells into cantrips for them.