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The IloCplex scripting class does support adding lazy constraints, see https://www.ibm.com/support/knowledgecenter/SSSA5P_12.10.0/ilog.odms.ide.help/refjsopl/html/IloCplex.html?view=kc#addLazyConstraints. So I think the answer on the respective forum should be corrected and the Aha at hand can be closed as "implemented".
Note that lazy constraints cannot be added to the opl model, but that is consistent with all the other APIs. Lazy constraints are not considered part of the model, they are special information to the solver. So it is fine that they can only be added to the solver.