You refer at many points to our "speed". What does that mean? With respect to what? The basic finding of relativity is that there is no special reference frame to measure speeds in. So you can take a whole batch of stuff, say everything in the vicinity of what later became our galaxy, and pick a reference frame where its average velocity is zero. Let's do that, for simplicity.
So now the question becomes simpler: was any of the universe far enough away at the time of last scattering for it to take until now for the light to just be reaching us? The answer is yes. There are a few reasons for this.
1. We have no compelling reason to think the universe is finite. If it's infinite, there's nothing more to explain.
2. If it were finite and small, then the light would still get here, just wrapping around the curved universe a few times on the way. That's not the situation in our particular universe.
3. If the universe were finite but big then the light from distant parts would get here without wrapping around. This is probably what puzzles you, since it would seem that so much more time has elapsed since the last scattering than before that nothing could have gotten far enough away. It turns out that things don't work out that way due to relativistic effects. Even ignoring general relativity (a bad idea for this problem) one can see why.
Say we pretend we can use a special relativistic reference frame. Look at some region that's young by its own local time, just reaching last scattering. If it's moving away from us at speed of almost c, then the relativistic time dilation says it's much older in our reference frame. It will have had time to get far enough away from us, according to us. The math is different in general relativity but that conclusion remains.
Mike W.
(published on 01/02/2013)