Personally, I'd like to see a Basic version of 5e that is more basic than "just remove feats, multiclassing, etc."
For example, with core mechanics, removing skills and bonus actions.
Interesting proposal...
I probably forget something, but on the top of my head these are bonus action abilities that are built into classes:
Barbarian: enter/exit Rage
Bard: use bardic inspiration
Druid: end wildshape
Fighter: second wind
Monk: extra unarmed strike, flurry of blows, patient defence, shadow step
Ranger: vanish
Rogue: cunning action
Sorcerer: flexible casting
The REAL deal of bonus actions is to be usable in the same turn as regular actions, but only one bonus action per turn. It gets complicated only when you have multiple abilities competing for that single bonus action per turn, but only the Monk has more than one.
A "basic" game can simply re-word the abilities above as "once per turn in addition to your action". Everything will be fine because there are no feats or multiclassing. Bonus action spells could be simply excluded from the basic game.
The Monk would need an extra rule saying that those four are mutually exclusive (group them together like the Rogue's Cunning Action).
There are some subclasses abilities that use bonus actions but actually not many. However I believe that a "basic" game would default each class to the simplest possible subclass so it should be possible to avoid those with more bonus action abilities.
So you wouldn't even need to restrict Basic to four classes like the actual 5e Starter Set does. I believe that a Basic game doesn't need to limit available races and classes, but rather to remove other choice points, including ability scores generation to keep everything as simple as possible during both the game and character creation/advancement.