I can't know why Amy Chua felt this way of course, but acting was considered naff just a couple of generations ago and maybe a very conservative person would still feel so? Acting could, for example, require a certain level of undress and most definitely intimate contact with the opposite sex. Instruments other than piano and violin might require an undignified stance and give fewer opportunities for solos. Just guesses.
I'm not sure how it works in the Chinese culture, but at least for a certain kind of conservative in the West it seems to not be it. I mean, the same people are ok with making a child to learn ballet, which is even more career limiting than acting and involves showing up in pretty tight outfits and getting pawed a lot. Or making a child learn a buttload of poetry by heart and making it recite it to other parents, and, frankly, unless you're going to be an actor or maybe a priest, exactly in which situation will you ever need that skill?
My subjective impression is still that the kinds of things they choose for their kids seems to coincide awfully conveniently with the set of stuff which:
1. Is easy to show off. You can make your kid recite poetry to someone or play the piano or whatever. But you can't really do a whole play at home each time you have guests, can you? Both for ease and time involved, the acting is just not easy to show off.
If you were the kind of praise addict making a kid grind and throw away their childhood to get you the daily shot lest you get the jitters, which would you choose?
2. is traditionally accepted as high class stuff. Most of these people don't really know any more _why_ and for what _purpose_ the people who could afford a piano would want to be able to play it (hint: they didn't have CD players to entertain the guests with) or recite poetry (ditto) or train to do elaborate dances (hint: that's how they danced at a party.) If they did, they'd just get the kid a CD player and get her to learn to dance disco. But they remained in the collective memory as stuff that the posh folks used to do.
By comparison, acting didn't use to be glamorous stuff until well into the 20'th century. The vast majority of artists were around the point where circus clowns are now. Poor, a dime a dozen, and getting less respect than Rodney Dangerfield
Plus it was not considered as intellectual and refined as, say, opera or ballet. So it wasn't really something a posh family would prepare their kids for.
So, yeah, wannabe snobs and defenders of the old values and ways, don't remember why it was like that, but they stick to it anyway.
ETA: and in case it's not blindingly obvious already, let me stress that the purpose was precisely to prepare people to do well at social occasions. It was the kind of skills that would prepare one to mingle and socialize at a party. The "chinese mom" turning it into a grind at the expense of depriving her children of any social contact or skills or friends, is a depressing way to miss the whole point.
