It all comes down to installation technique.

I think there is only one way to install the king metal components. Is there a method that is best?

My builder does iron balusters by default. I've been in dozens of their model homes and none of the stair railings has felt wobbly to me at all.

I'm guessing what would make them wobbly or not would not be the balusters, but the newels (and how they are attached). I've seen some pretty wobbly metal stair handrails, almost always due to a wobbly install at the newel location. If you have a good sized wooden newel post at each end of a handrail, I don't think it matters in the least what kind of other post you put between (cable, plexi, metal or wood). I don't think they provide much support--only protection ("a wall")

I was going to do a matching iron newel post, not a wooden one. Is this a mistake, strength-wise?

As others have said, newel anchorage is the key to a strong hand rail system that does not wobble. Your carpenters might struggle anchoring an iron newel post as compared to one that is wood. Just an opinion: the iron newel might be a bit too much of the same. It is the contrast that is so attractive with an iron balustrade with wood newels and wood handrail.

There are a few steps with railing at the bottom and then a balcony about 4-5 ft long at the top. Which profile wooden newel would look nice? I've not found one that matches aesthetically.

We have iron balusters and ours are tight (not wobbly) and we haven't had any problems out of them at all!

Here is a link that might be useful: Our home build/sell blog

You also have a wooden newel post.

Maybe that gives an idea of what the two look like together though, 3apples.

I don't think either run in your plan is long enough to be wobbly: it's a few steps at the bottom and one end will be wall anchored and the upper run is short and will also be anchored at the ends. You are also using a brass handrail so I think the whole thing is going to lock together.

Just for general information, the house in question is Georgian Revival. A house of this sort would have an all wood railing or an all metal railing, but generally not a hybrid. The wood iron combo is a contemporary version of Spanish.

"I think there is only one way to install the king metal components. Is there a method that is best?"

King makes many any styles.

Some are NOT allowed in residential (like the cable systems a child could climb or fit their head/body through).

Without more detail about EXACTLY what you are trying to do, no one can give you useful advice.

The best method for ANY newel post is to bolt it solidly to the side of a joist.
If a joist is not in the correct spot, blocking is added to create a mounting surface as deep as the joist.

Any type of toe-nailing, tow screwing, etc. or fastening to the floor or subfloor (no mater how many brackets you use) is markedly weaker.

You have a solid post 3-4 feet high making a nice lever to apply loading to the fasteners.

How much force will a 150 pound person create on the fasteners when they 'spin' around the newel holding with one hand?