The game isn't a stacked deck in that the outcome can be determined but there is a specific point in the game where you are guaranteed to only have a few tiles available (ex: 1024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 yields half the board) and you may or may not need a 2 to continue when the random selects a 4 locking your 2 in a corner or other similar manner.

I can't speak on statistics but I had best luck when following a set of heuristic rules that minimized the chances of locking valuable tiles in an unreachable location. If you stack the higher value blocks to a corner you're less likely to trap them with lower value blocks that are almost impossible to reach.

Ultimately I think the game is a better comparison to minesweeper where there are rules that will help you progress but at some point you're forced to guess and hope the outcome is in your favor.

Quick idea for empirically determining "winnability": implement the complement AI for computer moves [which are (i) position and (ii) value i.e. 2 or 4], perform a decent number of runs with depth limits for I.D. set much bigger, then look at the results. Also, I haven't looked at the code but I don't think looking into improved heuristics can hurt.

edit: I used 'look' 3 times in the last 20 words, what's wrong with me

I would go with Solitaire sine mine just lost. Did reach 1024 though.

Solitaire is perfectly winnable.

I have finished solitaire a lot of times.

I pressed solve from the start, and the AI lost.