"Hard" water primarily causes scale buildup, as well as changing the effectiveness of soap and other things that interact chemically with water.
In hard water areas if you like lather from your soap in the shower a water softener is necessary. If you like your white articles to not come out looking gray and dingy you need soft water. If you like long appliance life from your dishwasher, washing machine, and hot water heater, a water softener is necessary. And so on.
If you don't have very hard water it becomes less necessary. Still nice to have but a little tedious to maintain. But in very hard water areas, yeah, you want one.
Soft water isn't healthier (you could even argue that it's less healthy, but the difference is negligible, IMO).
But if your wife likes to drink tea, for example, having soft water is an understandable wish, because the tea tastes better when brewed with soft water.
If she's arguing fo a home-wide filter, not table-top:
She may reasonably wish for soft water, because you don't immediately see lots of water stains on the sink. And your washing machine and dishwasher live longer.
No, you certainly don't need soft water. But it's not quack science, it's actually solid science with the big question if it's worth the money to you.