Everyone has a balance of these things, but in my experience there are two extrema to approaching maths. Some people are happy to be solving the puzzles, and simply enjoy the thrill of the chase. Others insist on knowing why they should be learning this, and what it's good for.

Both are valid, but if there is A and B and you're naturally good at A, the natural tendency is to ignore or avoid B. But that then limits your box of tricks, and makes you one-dimensional. By all means take advantage of what you're good at, but work hard on the other aspects to enrich your abilities. Also, they cross-fertilise, so you get more than you might expect.

With regards your oldest, it's worth trying a bunch of stuff to see if you can work out what they prefer. Then do both, but be prepared for having to work harder on one aspect than the other. Your oldest might like puzzles for their own sake, or might insist on knowing what it's for. Explore both. Find maths in strange places, work through puzzles just for their own sake.

The critical key is to be doing stuff all the time.

Caveat: I don't have children, I have no formal training, but I do a lot of outreach, enrichment, and enhancement.