Dear Sir, Your understanding is not wrong. Mantras have overall effects too and you may carry on with it if you wish.
But you know what, my mother was a Heart patient and she used to visit Dr. C.K. Parekh, who was a Cardiologist (obviously). But all the same, he used to prescribe medicine for my intestinal infection and flu also…..very effectively. Does that ring a bell ?
Just for your information, I would like to share my understanding. Mahamrtyunjay Mantra, which is also known as Triyambaka Mantra is verse taken from Rigveda and dedicated to Lord Shiva. This mantra, in its basic meaning, is liberator from death (Mrytyormukshiya) and captivity of deadly disease (Urvarukmiva). Yes, it does bless for prosperity and well being (Pushtivardhanam), but all great mantras do bless for well being Mukesh ji. I am not suggesting you, that you are wrong but remember that specialist thing, I stated above.
My humble obeisances.