There are generally three accepted paths to power: Either you were born with it (innate talent, magic-in-the-blood, one of your parents was a monarch, etc.), or you earned it through training and effort, or it was bestowed upon you because of something you did. The fourth, often-not-accepted path to power, is bargaining for it.
Note that this is also seen in both religious hierarchies and aristocracies: Either you are born into a special role (e.g. you are the child of the previous prophet/monarch), or you studied and trained and passed some kind of test or won some kind of award through a display of that effort, or you were a great general or preacher or whatever and thus some authority vested you with the power as recognition of your skill. Or...you just bought you way into the position, whether through bribes or promises or alliances or whatever else.
So it seems to me that, if your goal is to collapse all of the arcane classes into one singular class that then splits by subclass, you should have one subclass to represent each of the above four paths: innate, trained, endowed, and bought.
You've got innate (Sorcerer) and trained (Wizard). That leaves endowed power and bought power. Warlock is obviously the latter, so this leaves open the question of what endowed power should be. And, for that, I think Bard actually makes a lot of sense. Both Bards and Wizards involve training, but there's a much greater emphasis on personal output for Bards; their magic is less a matter of gaining understanding, as it is about achieving great performance.
So why not have Bards be bestowed power by the College of Bards? They demand great performances in whatever field--poetry, song, dance, oratory, whatever--which is less about having technical skill, and more about adding something great to the College corpus. Greater magic requires enchanting more people--so the College benefits from having more Bards doing more performances in more places. Effectively, Bard magic is its own cause; it has power because people believe it has power, and people believe it has power because the College drives its members to always push the envelope of what performance can do.