You are quoting out of context.
The quote you present is from page 307, the following is from page 306
The situation shown in figure (a) corresponds to the case of central
forces, and we conclude that the conservation of angular momnetum
following from Newton's laws in the case of central force motion.
However, Newton's laws do not explicitly require forces to be central.
We must conclude that Newton's laws have no direct bearing on whether
or not angular momentum of an isolates system is conserved, since
these laws do not in themselves exclude the situation shown in figure
Newtonian mechanics comes with the implicit assumption that the effect of any force is independent of spatial orientation. If you grant that assumption then conservation of angular momentum is a logical consequence.
Obviously, if you decline granting one of the fundamental assumptions of newtonian mechanics then you have moved to a realm where newtonian mechanics may not apply.
As I understand it, Kleppner is arguing that we should allow for the possibility that the Newtonian assumption that implies conservation of angular momentum may not hold good.
Kleppner does assert that within its realm of applicability newtonian mechanics implies conservation of angular momentum.