Very little in the field of chemistry makes any sense in detail without quantum mechanics. If you want "phenomena which require quantum theories to explain them", just look around ... why is wood brown and leaves green and iodine yellow? What chemicals are stable versus unstable, why do different elements react different ways, why do salt crystals form cubes while ice forms hexagons?
None of these questions can be answered in a correct and consistent way except in the framework of quantum chemistry (and its consequences like orbital hybridization, delocalized electrons, resonance stabilization, Pauli exclusion principle, electron orbitals, the relation of light absorption and emission to electronic structure, etc. etc.)
These sorts of things are not usually discussed as motivation for quantum mechanics because it is a long and difficult road from the basic principles of quantum mechanics to explaining facts in chemistry like why ice is hexagonal. One might imagine that there is an alternative explanation of all the facts of chemistry that does not require quantum mechanics ... well, there isn't, but there is no particularly easy and pedagogical way to convince people of that. By comparison, there is a relatively simple path from basic principles of quantum mechanics to the two-slit experiment.
For the exact same reason, the conceptual breakthroughs in quantum mechanics did not historically arise from the attempt to explain why iodine is yellow. The path is too indirect. It required years of work AFTER quantum mechanics was established to understand how quantum mechanics was the only sensible explanation of almost everything in chemistry.