If walking is a result of the reaction to a kick backwards to the ground (reaction being friction), it appears that it should be true that the kick will have to be less than the kinetic friction (which is less than the static friction threshold = $\mu$mg) or the acceleration will never be more than $\mu$g. What happens if the kick exceeds this force, there still has to be a reaction? Is that reaction distinguished from the kinetic friction?
Depending on if you kick into the ground or place your foot firmly on the ground and then "kick" to move forward, you either have to kick with less force than the threshold values for kinetic or static friction respectively to maintain "normal" walking.
Overcoming this threshold value will result in your foot having a relative velocity to the ground, or in laymans terms, you will slip and slide. This does not mean you can not maintain walking, you will just not be able get a firm foothold. When sliding like this, the friction you experience is the kinetic friction threshold value, meaning you will not be able to use more force to accelerate your body than this value. This for example happens when you thrust too hard while walking on ice.
the kick will have to be less than the kinetic friction (which is less than the static friction threshold...)
Not quite. The kick will have to apply less force than the static friction. There is no need to consider kinetic friction at all. Only when the kick applies a larger force than the static friction limit, will the foot start to slide and kinetic friction will be relevant.
Those two kinds of friction are for two different cases:
- Static friction when there is no slip: $f_s\leq \mu_s n$
- Kinetic friction when there is slip (sliding): $f_k=\mu_k n$
$n$ is the normal force. Kinetic friction will only happen when there is (relative) motion; static friction will only happen when there is not.
So, you put your foot on the ground. It doesn't slip, so we are talking static friction. As you see from the formula, static friction $f_s$ can take any value up to a certain limit! Static friction will always have the magnitude that is needed to keep your foot still, but if it is not able to, because the required amount exceeds the limit, then static friction cannot hold your foot still anymore and then it will slip and start to slide.
Then you are in the kinetic friction region. As you see from the formula, there is no "limit" for kinetic friction $f_k$ simply because it doesn't vary like static friction can. It always has a specific value - just only during relative motion (sliding). This means that while you walk, your foot skids and slides backwards, but you still get some forward drift. Like walking in sand. Or like accelerating too much in a car, so the tires start burning because they skid and roll uncontrollably on the asphalt without grip but the car still begins to move slowly forwards.
The extreme case is a perfectly smooth icy surface, where $\mu_k=0$, so there will never be any friction nomatter what and thus never any forward drift. But this is only a thought-experiment as an ideal model. Ice in the real world does have friction - otherwise ice-hockey would be impossible.