I strongly agree use OEM parts only. Changing pilots or TCs can change the dropout rate and void the warranty & listing.
What you are describing here could be sediment in the base of the tank. As sludge builds up, it insulates the tank requiring more and more BTUs to heat the same water. Flush the tank PMI and see if that helps. The entire control can be replaced, which may be ok for a newer tank but if its about 7+ yrs old or especially if the anode rod has never been replaced, you may be better off buying a whole new tank. These will be more expensive because they have the Flammable Vapor Ignition Resistant design but they are also better insulated and more efficient.
BTW, when's the last time anyone inspected the vent connector or chimney?
Keep the fire in the fireplace.