telescope - What should I look for in an astronomy tripod?

I need to upgrade my tripod legs.

Yes, this seems to be the case

I don't need to upgrade the head, since it's rated to twice the capacity of the legs, and to twice the weight of the biggest binoculars I'll use.

Seems right to me. However, if you have some money to spare, I think you should consider investing in a Parallelogram Mount (P-mount). These usually look like long arms with counterweights and a place to attach your bino. They make it easier to adjust the height, observe objects close to zenith and are usable from a chair.

I should look for legs that are rated to twice the weight of the biggest binoculars I'll use.

I don't have this problem with binoculars as the ones I own are pretty light but with photographic equipment, I try not to exceed half the declared capacity. The better the brand, the closer you can get, I suppose.

I should look for a set of wooden legs, since these have greater stability than aluminium legs while being affordable, unlike carbon fiber.

I've used wooden legs for a small telescope and I wouldn't like to do it again. Unless you're only planning to use a tripod around the house. Wooden tripods are usually heavy, clunky and not very portable.

Carbon fiber tripods, on the other hand, tend to be very light so you'll have to be a little more careful with a big and heavy bino than in case of an aluminium or wooden one. If you get one, make sure it has a hook under the column, on which you can hang something to shift the center of mass closer to the ground.

I once had a carbon tripod almost tip over after a strong gust of wind when I had a large-ish telephoto lens mounted on it. Managed to catch it just in the nick of time.