I wonder what would have happened if you did put new sand in, and then ran the system. Not that I recommend it--smelly messy, job! We have (properly? I think!) maintained a sand filter for 40+ years. The sand we replaced around the 30 yr mark did not significantly change the pool function in any way. We had replumbed the pool and just thought maybe we should. Both before and after we backwash about 150-200 gallons once or twice a week. The only chemicals I've routinely used are chlorine, calcium, baking soda, and CYA. I haven't had much success trying aluminum based flocculants and filter cleaners. I've not experimented with adding DE to the filter. In your case, you had to do something, either change the sand or upgrade the filter.
A sand filter probably doesn't have the clarity of more modern systems--storms and high bather load will overwhelm it pretty quickly. If the tank wasn't sized properly, or if previous owners threw chemical willy-nilly into the skimmer instead of into the pool (calcium!) then it would explain your issues. I might also mention that my pool stays much clearer in colder weather than in warm. Your indoor pool likely provides a better temperature for algae/bacterial growth, 24/7, and the sand filter wasn't able to keep up. Your cartridge filter sounds great: if we didn't have water turbidity issues, plus a seasonal outdoor pool, I would be inspired to change over.
The strength of the sand filter is that it is simple, cheap, and will last a long, long time. The weakness is that it requires vigilant attention to FC, tank pressure, and it can easily be overwhelmed. The takeaway from this thread is that sand DOES need to be changed when you can visually see a problem in the tank.