One way of thinking about altitude is how many mistakes you have time to recover from before hitting the ground. The higher you are the safer you are - you haven't crashed until you hit the ground.
For a single-engine plane, 500' AGL depending on the plane and wind is enough to make a 180° turn and land on the takeoff runway if its clear. 1000' AGL will let you do it on just about any suitable surface at the airport - other runways, taxiways, grass clearings adjacent to them, etc - and give you a few seconds to be confused before deciding that you need to make an emergency landing.
Most birds fly quite low, and drone-based bird abatement at airports would be a safety win if done right (DO-178-B style software quality), but I've heard of bird strikes above 20,000', and been taught to expect that you're never flying above _all_ the birds.
Edited to reply directly to your 3000' comment like I meant to: At 3000' AGL, you can almost always glide to a safe landing point, unless you're over mountains or a dense urban area. But unless you happen to already be near an airport, you probably won't be able to glide to one. Glide ratios much higher than 10:1 are uncommon, so thats 5.7 statute miles. Lots of places are more than 5.7 miles from an airport.