Yes, guitar string metals really do matter
When you start experimenting with strings made of different alloys, weird noises might happen.
I can say with absolute certainty that when a guitar string manufacturer says a certain string is made of something different, it's not just lip service. Different metals do result in a different sound. Sometimes this is good, sometimes not.
I was able to wrangle a fairly decent sound out of these:
...but it comes at a slight cost, and I'm not talking about price.
This is a direct quote from the Ernie Ball page on the Extra Slinky Cobalt strings:
Ernie Ball Cobalt Slinky Electric Guitar Strings provide an extended dynamic range, incredible harmonic response, increased low end, and crisp, clear highs. Cobalt provides a stronger magnetic relationship between pickups and strings than any other alloy previously available. Cobalt Slinky guitar strings are also soft and silky to the touch, making string bending a breeze.
Is all this true? Yes. However, a drawback is that the increased harmonic response can in fact be too good.
On my Squier Stratocaster, when I palm mute the A or D strings, really high harmonic "ping" noises can be heard. Problem with the pickups? No. Problem with the guitar? No. What's happening is that since the response of the strings is so prominent, it brings out unwanted harmonics for certain play styles.
Is this a problem with the string? No. The string is actually doing exactly what Ernie Ball said it would... a little too well.
This can also happen with stainless steel strings
The go-to example for this are D'Addario XL ProSteels. On that particular model, the wound strings use a stainless steel wrap instead of nickel plated. This string is not the same as the Ernie Ball Extra Slinky Cobalt, but like the Cobalt strings does change how things sound.
With ProSteels, yes, they are brighter. Are they too bright? That I don't know. I haven't tried them personally but might in the future.
What I do know is that whenever you use a string where the metal material is not the standard nickel-plated stuff, the sound does change. Sometimes the change is good, not-so good, or introduces a particular quirk.
Can changing to a string with a different metal "save" a pickup?
Short answer: No.
Using a string like the Cobalt or the ProSteels can and does result in a better magnetic attraction from string to pickup. However, even though this is true, a string can't magically revive a pickup magnet that's lifeless or close to dead.
If you have a pickup that sounds great but is otherwise low-output (some humbuckers with alnico II magnets are like that) and need something to give it a boost in harmonic range, that's a solid reason to use strings with different metals in them...
...but just be aware that sometimes changing to a string made with different metals may bring about weird/unwanted sounds. If it does, there's nothing wrong with your guitar nor the string.