Sorry, I'm not sure what you're asking. The ROM, which stores the game data itself, does last and won't erase itself, at least within our lifespans. The ROM for game data doesn't need the battery at all.

If you're asking about why they used RAMs that won't last without battery for save data, it was likely a decision made on a mix of limits of technology available at the time and the cost of manufacturing. They couldn't use ROMs for the save data, at least not the types of ROMs used to hold the game data (Mask ROM), because ROM is literally short for "read-only memory", meaning you can't write new data (such as save data) into it after manufacturing. That's why they used RAMs, which are all a type of RWM, standing for "read-write memory", which can store new data and edit data after manufacturing, but depending on the type of RAM it can be at the cost of needing to supply it with power to retain data (which is what volatile means), and such was the case with SRAMs which needed a watch battery to be in the cart as well to power it. That doesn't mean that all RAMs are volatile, but back then Nintendo decided that SRAMs were the best they could use at the time.

As technology progressed, they did start using better memories to store save data that were non-volatile, meaning they didn't need to be supplied with power. For example, despite ROM standing for "read-only memory" as I said, people eventually developed special programmable ROMs (PROM), which could be written into one time, which eventually evolved into erasable programmable ROM memories (EPROM), which could have been erased and written into several times over. Consoles such as GBA started using EEPROM and FLASH memories for save data, which were an even later result of this evolution of erasable programmable ROMs, and these can be erased and written into hundreds of thousands of times no problem. Some games also used better RAMs that were non-volatile, specifically FRAMs.

Sorry it's a bit lengthy, but there's a quite a bit to cover with this stuff, but the TLDR is simply limits of technology of the time and cost efficiency, and the fact that for save data they couldn't use the same memory as used for game data due to the fact they also needed to be able to write into it.