I looked at a piano that the seller claimed had a soft close fallboard (Kawai) but it just came right down. I don't believe him that it's supposed to work that way. Assuming it's broken, is that something that can be repaired?

What do you mean by it just came right down? Did it simply slam shut with a bang - or did it go down slowly with some resistance ?? It's either slow close or it isn't - or perhaps the owner doesn't have a clue.
And yes, if it's broken, it can be repaired - but it probably isn't worth the effort to do so. Slow close is nice - but it isn't essential.

Sorry if I wasn't clear - it didn't come down slowly, but it didn't close with a bang either. It felt like it came down at the regular pace but weighed less, if that makes sense, which is why I am guessing that it does have the feature but it isn't working. And I doubt the owner knew much about it, the piano also hadn't been tuned in 5 years.

It might be worth it to me to fix it for our kids' sake - are you saying that it would be very expensive to do so?

Slow close fallboards work a variety of ways. It is possible it is working, but simply closes faster than another system you have seen.

Good news - the slow close mechanisms are pretty much in stock for most manufacturers because they do fail, especially when a child (or adult) tries to force them to close quicker than they are designed to. This breaks the seal on the cylinder that operates it and makes replacement a neccesity.

Good Luck,