Be careful relying on studies in this area. The "growth mindset", like much of psychology over the last several decades, is subject to the replication crisis.
At the very least, the size of the effect is almost certainly quite small.
I am not sure there is one sure fire way to teach kids. Each one is an individual. What works on one child may fail horribly in another. I think that is what we ultimately try to ignore. We want to simplify everything and say we need x or y. When in reality its that this kid needs x and that child y. But doing that is more work, people are lazy. Always looking for instant gratification. I know I have fallen victim to that myself.
I you watch the TED talk by Angela Duckworth, author of Grit book, she says one thing where they still really have not figured out how to teach resilience to kids.
I was speaking to a high school teacher last week, and they use these grit scores and all the tips mentioned in the grit book. But at the end of the day, they still do not have a sure fire method to teach it or the growth mindset.
Years ago I spoke to a retired mathematics professor on this subject. I remember he pointed out that Évariste_Galois was an interesting case in home education. There was not too much written on this, but he was taught by his mother the classics. What he accomplished at such a young age is still be used today in all sorts of fields.
Kids who went to Sudbury Schools (no curriculum, democratic free schools) report developing high resilience. In these schools kids do what they like: read, videogames, singing, whatever, without any academic pressure or expectations. They try stuff and fail. They have to think about what they actually want to do, rather than looking to the teacher to tell them what to do.
A great book with a similar perspective is Summerhill by A. S. Neil.
We do Sudbury inspired home schooling. My daughter learned to read "late" by most standards (at age 9), but once she decided to it only took her a few weeks to be at the same level as her friends who go to public school. She also learned to read 3 languages at once while her friends can only still speak just one language (part of our home schooling has been language immersion). I feel proud of what she knows, but most proud of her motivation and determination to learn what she wants to.
I think the point is that intelligence is a tool. By itself it doesn't get you terribly far (assuming you want to get somewhere). You need to apply it to something diligently. In other words, intelligence is necessary but not sufficient.
Kids will value what you compliment. If you compliment their smarts, they will simply work towards always being "the smartest guy in the room", and making everyone else in that room constantly aware of this in the most obnoxious way they can come up with. If you compliment their work they'll strive to get recognition for that instead, which is a way more functional attitude in the longer run!
Intelligence is like physical beauty. It probably wouldn't be terribly productive to tell your kids all the time how beautiful they are even though attractive people tend to be more successful in life.
Why not compliment both so they know it's an important combination and to take advantage of natural gifts? Breaking rocks is hard work but that is not how I'd want my gifted child spending their time. I'm not saying kids should be showered with compliments either but if my kid genuinely surprises me with their wit I'm not going to be such a stiff piece of shit to think I'm doing them a favor by not complimentin them.
What, precisely are you cautioning against? Merely against folks who excessively bristle at "you're so smart?" Or are you cautioning against the growth mindset in general?
I only happened upon the growth mindset in the past year or so, and it's changed (or is changing) my life. I don't see how it's not possible that it's not profoundly true. That doesn't mean that you have to use one author's books or terminologies, but the point of it is a relatively profound truth. It resonates so strongly with me, and explains so many things about my anxieties and emotions, as it apparently does with others. And there's nobody pushing the opposite.
At the very least, the size of the effect is almost certainly quite small.