FFIII NES has gotten a lot better over the last 2 decades due to mods. However, the FFV spell system still beats it in terms of convenience for how spell learning and use works.
In FFV, once the party learns a spell (like Fire or Cure), everyone in the party knows it permanently. (Each party member can only cast it if they otherwise meet all prerequisites, such as being the right class, having enough levels, having enough MP, and being alive but not silenced.)
In FFIII NES, each character normally needs to learn a separate spell copy, but can only have 3 total spells per spell level active at a time. Thus, finding a Cure2 spell means that only one character can have it equipped at a time, even if that character can't cast that spell such as due to class or level.
What I propose - and this requires a feasibility check for implementation - is that spells (Aero, Mini, Toad, etc.) are kept on a separate inventory screen and sorted by type - Black, Summon, White, etc. - and each character picks spells from that screen to learn based on spells found. For example, if the party found the Cure2 spell, anyone could prepare that spell (that is, put it in a spell slot of that spell level) and cast it as normal, instead of needing to find 4 copies and handle spells as if they were items, competing with the inventory space best left for weapons, armor, and consumables.
In this model, spells would not be sellable to vendors nor trashable, but held onto forever in inventory. However, due to likely technical limitations of the 3 spell slots per spell level setup, players would still need to swap in whatever spells they wanted active. (If these limitations were altered, then the number of spells of each spell level each character could prepare would be the new limit of this mod.) In general, my preference is to make spell preparation as easy as changing classes (and having access to all spells castable by that new class), thereby coming as close as possible to FFV.