Titanium was useful when we needed very high strength on a very large part - a good example is the trunnions that attach payloads to the space shuttle cargo bay. These are large metal bars a little over 3 inches in diameter and maybe a couple feet in length. Aluminum is not strong enough. Steel would be punishingly heavy. (do the math - these things are beasts). So titanium can be a good fit for stuff like that.
But most steel parts were small - things like bolts, rod ends an the like. They tend to be specialized with very specific detailed requirements for things like fatigue, frictional characteristics, and that sort of thing. Because they are small, the weight just isn't that big an issue. Bigger stuff was almost always aluminum, which is actually a lot stronger than most folks realize. At least, the good stuff is.
Titanium also has some other odd properties that make it troublesome in space. I wasn't a materials guy, but they were always bitching about "hydrogen embrittlement" and the fact that large titanium parts do not burn up on re-entry like aluminum does, which presents a debris hazard to people on the ground. (I don't know how big a problem that really is, but people didn't want to be dumping space junk on top of people's heads. Honestly, I just took this on faith - people were concerned about it, but it always struck me as odd.)
I'm glad to have 6061 on my bike instead.
Thanks!
Now suppose instead you were trying to traction the wire (pull it). There's no structural change that will make it stiffer -- in this case all you may want is that it is very strong -- in which case you simply go for the best strength/weight or just pure strength.
How are you defining strength? Tensile strength?
If you went for a compression stress however you'd probably want that same bending stiffness (by making a hollow tube) because rarely the force is perfectly axial.
The idea is that design is material dependent. You can't assume you can just swap out parts for lighter/stronger/stiffer parts and still have everything work.