Unfortunately I don't have the time to do this right now, but I still want to through this idea of how to tackel this (very hard) problem out there.

This method somewhat resembles what is often used in statistical physics. Cf as well to ergodic theory and the likes (we would have to prove ergodicity.... but well...).

For every field store an array of probabilities that this particular field is in the state 0 (unoccupied), 1 (has the number $2^1$), 2 (number $2^2$), ..., 11 (winning field $2^{11} = 2048$).

Let $P(i,j,x)$ denote the chance that the field with coordinates $(i,j)$ is in state $0\le x \le 11$.

Each move (up, left, right, down) has a clearly defined effect that can be expressed as a set of rules for example
$$P'(i,j,x) = \underbrace{P(i,j,x)}_{\text{already was in this state}} - \underbrace{P(i,j,x) * (P(i+1,j,x)+P(i+1,j,x))}_{\text{leaves this state due to a join or move to the right}} + \underbrace{P(i,j,x-1)*P(i-1,j,x-1)}_{\text{joines this state due to join from the left}} + \dots
$$
where $\dots$ stands for the more lengthy terms due to moving tiles (eg. somewhere to the right is an empty tile and left of me was the value $x$ in the last step).

The insertion of random numbers will simply decrease the likelihood that each field is unoccupied and respectively increase the state's $x=1$ and $x=2$ chances
$$ P'(i,j,1) =P(i,j,1) + P(i,j,0)/16 \\
P'(i,j,2)=P(i,j,2) + P(i,j,0)/16 \\
P'(i,j,0)=P(i,j,0)*\frac{14}{16}\,.
$$

Any move and the random insertion step are alternated on our current state (the vectors of probabilities). The likelihood of loosing in the current step is equal to the likelihood of all fields being occupied before the random insertions step. The likelihood of winning is the likelihood of any field being in state $x=11$ after the chosen move.*

Clearly the probabilities *should* be correlated. Assuming that they are not is somewhat equal to the "molecular" chaos that is assumed in statistical physics / ergodic theory. But, assuming that we are indeed ergodic with this description of the model, we can get the chance of winning the game after $n$ predefined steps (and not loosing it before) by iterating this $n$ times. This way one could compare different strategies easily, but would still have to test several random chains of moves to get a decent average. (We only implicitly averaged over all possible positions of the inserted $2$ and $4$ fields)

(*) Note that we have to remove any winning states from our vector of possibilities before each random insertion. Clearly we did not win yet if we are still playing. (Also this is necessary to have any chance at being ergodic)