I think 2000ft is the most suitable atl to deploy landing gear, we still have time to check if they are fully down or not and feel how it affects flight control
The Shuttle is a 100 ton flying brick, and you have one shot at a decent landing. There's no point in checking whether the gear deploys or not, because if it does not you have to land anyway, there is no go-around procedure with something that glides as badly (besides, if the gear doesn't respond to hydraulics within a fraction of a second, it's deployed by pyrotechnics).
In addition, it's a pretty solid drag device, so once you deploy gear, you will lose lots of energy. If you start flaring low on energy, you will get into trouble because you can't pull the nose up more than 15 deg to controll sinkrate otherwise you will scrape the tail on the ground. So an early gear deploy schedule will mean that you have to first deploy speedbrake, then deploy gear, then retract speedbrake to account for the additional drag, then re-deploy speedbrake during the final flare. A late gear deployment schedule is far less hectic, you can basically keep speedbrake settings all the way to touchdown.
Landing a spacecraft is somewhat different from landing a GA aircraft - and NASA had good reasons to design the procedure such that the gear is deployed late (unless you have a speedbrake stuck in retracted position, in which case you would deploy it early).