This analysis falls short if you apply three body orbital mechanics. Going barely faster or slower than the Earth in close proximity to Earth will slowly cause

This analysis falls short if you apply three body orbital mechanics. Going barely faster or slower than the Earth in close proximity to Earth will slowly cause Earth's gravity to pull the spacecraft back in. This can mean all sorts of funny shaped trajectories (horseshoe asteroids etc). You need about 11-12 km/s to relative to the Earth to escape Earth's gravitational pull from LEO (only a bit more than it takes to reach the moon). Earth itself is orbiting the sun at about 30 km/s. To not get re-captured, the spacecraft needs to be far enough in distance or the difference in velocity needs to be large enough. How much is "enough" depends on the time scale, but for spacecraft we're talking several km/s.