Deriving rate law from a reaction mechanism for a chain reaction using steady-state approximation
In the book that Wikipedia cites (Advanced Organic Chemistry: Reaction Mechanisms
By Reinhard Bruckner, ISBN 9780080498805), they have a different set of assumptions. They are saying the formation of chlorine radicals proceeds by a fast equilibrium. Then, they say that the organic radical is at steady state, ignoring reaction (4) given by the OP (the termination reaction).


Where am I going wrong?

I'm not sure why your steady state approximation is less appropriate than that of the textbook. I'm sure that there is some experimental data (and some fitted rate constants) that would speak to that. Also, the appropriate approximation depends on the initial conditions.

[Comment from Poutnik] The initiation rate may not depend on [Cl2]( for high enough [Cl2]), but on irradiation level only. i.e being a reaction of zero order.

The derivation in the book is for the thermal reaction, but the OP asked about the photocatalyzed reaction, so there is a discrepancy there.