Depending on the application, MANY aluminum alloys can "compete" with steel. Especially if weight is limited, but space is not, and you're worried about buckling or bending. OTOH, if the load is tensile, the part is necessarily stubby enough not to collapse, or the temperature is high, a good grade of steel will be hard to beat. Of course, if you are constrained to 1/8", steel is your best bet.
As per your edit, which I saw just now, an aluminum rod that's 3/16" in diameter will be significantly stiffer than the 1/8" steel rod.
BTW, one can straighten music wire. For 1/8", you'd need a piece that's somewhat longer than you need, a way to grab the ends securely, and a way to apply a bit over a ton and a half of tension*. ;-) It works great if you're straightening .030" music wire, anyway, and I can do that without mechanical assistance. It sure feels weird when the wire starts to yield.
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*On further thought, applying that much force wouldn't be so hard. I own a come along that could do it. Like this:
https://en.wikipedia.org/wiki/Come-along I'd need lots of extra music wire at the ends to wrap around things, though. Or appropriate fittings and a torch to braze them on with. It would be important to stay out of line with the wire, in case something broke.