The periapsis will always pass through where the collision happened.

To circularize at a higher orbit you would need secondary collisions on the other side of the earth.

You're right that all the fragments will pass roughly through the impact point in orbit. But it's not always the periapsis.

1. The normal or anti-normal delta-v imparted by the explosion/fragmentation (i.e, the velocity imparted perpendicular the plane of initial orbit) will cause the orbital plane of the fragment to change. The new orbit will intersect the old orbit at the impact point. Meanwhile, the eccentricity (the stretch of the orbit), semi-major axis (the size of the orbit) and displacement of periapsis from the impact point (the orientation of the orbit) remains the same as the initial orbit.

2. The prograde and retrograde delta-v (velocity imparted tangential to the orbit) will cause the diametrically opposite side of the orbit to rise or fall respectively. Here too, the new orbit intersects the old orbit at the point of impact. But since the impact point isn't guaranteed to be the periapsis or apoapsis, the above mentioned diametrically-opposing point also cannot be guaranteed to be an apsis.

3. The radial and anti-radial delta-v (this is in the third perpendicular axis) will cause the orbit of the fragment to either dip or rise radially at the point of impact. Again the impact point remains the same for the new orbit. So the new orbit will intersect the old orbit either from the top or the bottom. The new orbit will look like the old orbit with one side lowered and the other side raised about the impact point.

So none of three components of delta-v shifts the orbit from the impact point. You can extrapolate this to all the fragments and you'll see that they will all pass through the impact point. The highest chance of recontact exists there. However the perturbation forces do disperse the crossing point (the original impact point) to a larger volume over time.