We can only conjecture that the reason has to do with high levels of calcium, magnesium or other metal cations and therefore that the water must be hard. We don’t know what the levels in the well water are and we don’t know what the levels in the softened water are. Answering those questions is simply impossible.
We can know the tests for GH/KH of the softwater very soon (per test kit), and can know the well amounts of GH/KH if the extra faucet is added.
Are you implying that it is "danged if I do, danged if I don't" since either I am using softened water which has bad qualities or hard water with "calcium, magnesium or other metal cations" impurities?
If YES, then is the only remaining solution a ~$150 RO/DI system? This would also need to be stored into large containers because of the amount needed and the slow production time.
If YES to the RO/DI, then would I still need to use the well for RO/DI or would the softened water be OK to use?