Why is the TinyMCE Editor Toolbar Different for Mosaic Tiles than in General Page Editing?

Yes, all built-in TinyMCE buttons and plugins should be addable like that. There is also support for configuring the context menu.

Why it is different in the first place is a long story. Legacy of many Deco sprints before Mosaic. It was designed well before Plone 5 TinyMCE and it was originally supposed to replace "the default TinyMCE" altogether. Also, there was supposed to be control panel for ease configuration of those actions, but it was removed when the configuration was made even more flexbile (to make it possible for each tile to have different/optimized TinyMCE configuration).

Example of code-button:

Examples of contextmenu: