That's because the four velocity is not really a velocity. The path of a particle through spacetime can be described by a function p(r) = (x(r),y(r),z(r),t(r)) that gives the position (x,y,z,t) as a function of a parameter r. This way the point p(r) traces out some path in spacetime.

Now, the same path is described by many different parameterizations. For example, if we pick q(r) = p(2r) that's the same path. The point varies twice as fast with r, but that has no physical meaning. The total path that is traced out in spacetime is still the same path. That is, the set {p(r) | r in [-∞,∞]} is the same as the set {q(r) | r in [-∞,∞]}, and it is only this set that has physical meaning.

In fact, we can also pick q(r) = p(r^3 + r) or some other parameter transformation.

We might look at the vector p'(r) tangent to the path. The length of this vector depends on the parameterization, but its direction doesn't. So p'(r)/|p'(r)| is a true physical property of the path, but the length |p'(r)| is just an artefact of the parameterization that we chose to describe the path.

In order to make the parameterization somewhat more canonical, we usually pick a parameterization where |p'(r)| = c. As it happens, the four velocity is defined to be p'(r) given this choice of parameterization.

So the length of the four velocity is c simply by definition. This statement has no physical meaning. The statement "we travel through time with the speed of light" is uninteresting. It tells you nothing about physics. It only tells you something about the conventions we use when we describe paths through spacetime. We could very well have picked the convention |p'(r)| = c/2, and then we'd have the statement "we travel through time with half the speed of light", or even |p'(r)| = 2c, and then we'd "travel through time with double the speed of light".

Given any path through spacetime whatsoever, we can pick a parameterization such that it travels through spacetime at the speed of light. So the condition that something travels through spacetime with the speed of light places no constraints on the path that it takes. The statement "we travel through spacetime with the speed of light" makes it sound like there are paths that do travel through spacetime with the speed of light, and there are paths that do not travel through spacetime with the speed of light, and physical particles go along paths of the first kind. This is wrong. There are no paths that do not travel through spacetime at the speed of light, because whether or not it does is a property of the parameterization that we choose to describe the path, not a property of the path itself.