While some fish are very tolerant of water conditions, they are not infinitely adaptable, they are typically tuned to a limited window of conditions that includes salt concentrations (ionic strength), specific ion and organic compound concentrations (some heavy metals and ions are very toxic to fish), pH, and oxygen concentration, not to mention temperature and pressure. So make sure to consult with an aquarium specialist (perhaps a book or internet site will do) to find the right conditions for your fish.
Cold blooded animals such as fish cannot regulate their body temperature to any significant extent, and as metabolic processes tend to be tuned to a narrow temperature window, fish do best within that window, and tend to die outside of it.
FYI I killed my first pet fish (a goldfish) within 24h after placing it in untreated tap water. I got very upset and blamed my mom.
Back to topic: gas solubility typically decreases at higher temperature, as you rightly point out, and this can be a problem for instance in hot summers, often also due to a combination with other temperature-related factors. As you have had an aquarium for over a year, and presumably your fish have survived that long, they seem to be reasonably hardy cold water species (depending where you live) and I don't see a good reason to regulate the temperature unless something is expected to change such as a significant drop in the room temperature. However I'd prefer not to get involved in a dispute with your mom!