The emptiness of the deserts

There are no know problems with deserts. You could use a software like Chunkbase to find the nearest one. Otherwise, keep exploring!

It's entirely possible - the biome map is generated with accordance with temperature map, which often spawns very large distances. You may see thousands of blocks of taiga, snow plains, ice spikes, frozen ocean, extreme hills, snowy taiga, mountains, giant tree taiga, with occasional birch forest, and not a sniff of a desert, a jungle, a mesa or anything like that, because the whole expanse is determined 'cold' and only generates cold biomes. It's very rare to see a cold biome neighbor a warm biome. Or you can travel for thousands of blocks and never see a splotch of snow outside tips of mountains.

2000 blocks isn't much. Scout further, exploring mostly to warmer biomes - savanna, mesa, jungle, swamp, plains, regular (mixed/oak) forests.

This is a map of a server, nothing very special about the seed choice. The area is about 10,000 x 10,000. Observe the upper-right quadrant of the map. That's a 5000x5000 area without a single desert. Lower left also has hardly any. Upper left - savannas, deserts, mesa, and just a splotch of snow in the far corner.