Tinkercad is an exceptionally powerful tool that is amazingly intuitive and incredibly easy to use. It has the ability to do even more with just a few more tools that are present in many modern CAD programs and would increase it's functionality exponentially without cluttering it up or making it too unwieldy for the beginner or student.
First: LAYERS!! Currently it has a 2 layer system, Visible and Invisible, where you can isolate elements for selection and visibility. Most CAD programs have the ability to create layers and assign elements to these layers and the ability to do so with more than just the Visible/Invisible layers would be incredibly useful. Specific elements or assemblies could be grouped together on a layer as individual elements and made visible or invisible at a single click. We would be able to switch elements off so we could work on specific parts of a drawing with the visible/invisible feature only affecting those parts we were working on rather than having to repeatedly go back and make elements we aren't working on invisible. Right now everything is either on or off; Layers would give the user the power to select just WHAT was on or off at any given time.
Second: Alignment points! Once an object is grouped, you can only align it to other elements by left/right/center. Perhaps during the grouping operation you could assign an alignment point so that different grouped assemblies could be aligned according to their alignment points, thus aiding in mating parts together.
Third: Align to Center or Outside! It would be nice to align the center or outside edge of one object to the edge of another. Perhaps when hovering over the current alignment dot, a sub triple-dot "menu" appears allowing the object to be aligned left/right/center of the current dot you are currently aligning to. Currently we are only able to align inside an object rather than along or outside. This would also allow two objects to be butted up against another by alignment rather than having to create a workplane on the object to be aligned to and "dropping" the aligned object to that workplane.