I looks like the new block appears randomly. If the new block appeared even if you didn't move anything it would be easy, but they don't and that can ruin your strategy.

I got to a 512 block in a corner and then got blocked by a new block in the same corner.

I think the strategy is very simple. Pick a corner and mash keys that move into that corner. For example for bottom-right corner, just mash right and bottom. When you can't move anything anymore, move into the direction that keep your highest block in a corner and then continue to mash and hope that a new block doesn't appear anywhere close your highest block. I have gotten a 512 block in 2 minutes twice( score over 6000 ). It would only take time to reach more, and maybe actually look at the board instead of mashing; but that is problematic since the new block appears to be random.

1024 could be reached in under an hour and 2048 in a day, i think.

It would be better if you got the position of the new block. That way the game would be solvable in some normal time.

I tried a "tumbling" strategy, where I would press right, down, left, up, repeating. You'll see that the blocks end up tumbling around. I was able to get 256 several times.

I also tried a very deliberate (and slow) strategy of actively trying to build up the necessary matching blocks and was able to get to 512 with a score of 5900.

I seem to pretty much always get a higher score by button mashing, or going up down left right in rotation than by trying to figure it out. :(

So far the best heuristic I've found is to not let smaller blocks get trapped behind bigger ones.