RO system recommendations

Looking for help with an RO system, if it's even possible to get what I want.

1. I'm on a well. The unfiltered well water tastes almost kinda salty, and smells like change sometimes so I know there's probably iron in it.

2. I'm getting a softener installed, but soft water tastes just as bad.

3. I want the RO system to feed my fridge, not a faucet near the sink. We all like our water ice cold, and the fridge we bought has the automatic filling pitcher in the door. I plan to bypass the fridge filter.

What systems actually support connecting to a fridge? I read that this is an issue with some systems, not an issue with others, but there's not a lot of clarity.

Is remineralization a gimmick? If not, is it just trial-and-error to find a remineralizing filter that tastes good to us, or is there a method to the madness?

Also, and this is mostly just to make sure, I planned to feed the RO with softened water, but is there any chance this is a bad idea? Wouldn't be too difficult to run a 3/8" line from before the softener if it makes sense.