I wonder if, instead of new blocks randomly showing up after you make a move, if they just showed up after a period of time. If a new block showed up right on top of an old one some kind of rule would take effect, like add them together, or the larger wins or something. As "levels" progressed, the rate these new blocks showed up would increase. The goal would probably have to be changed, instead of just getting to 2048, get to, or exceed, or come in just under some arbitrary number for the level or something.
Might add new strategy elements to the basic idea. You might not want to make a move, hoping the next blocks add to a block that's already on the board...maybe even "hint" where the next blocks will appear, but not the value. Tetris after all shows you the next block that will appear.