I have a foggy memory of rewinding a tweeter decades ago. I determined the gauge, then checked the resistance of a good tweeter to find out how many feet it had.
As has been said, I'd check each speaker element. A battery would work, but I'd check resistance with a multimeter. If it looked good, I'd clip the leads on and see what happened when I pushed the cone.
Ten years ago, I thought something was wrong with my CD player or my amp. Then I discovered that the foam surrounds on 6 woofers and 4 midranges were rotting. I found a place that sold repair kits.
In the last few years, I've been thinking of complaining to the FCC that lots of FM stations are broadcasting distortion. Then I discovered that my electrolytic crossover capacitors were bad. Some estimate that electrolytics last 5 years on the shelf. Mine were 30 years old. A month ago I replaced 18.
The ones I took out were 10% precision electrolytics. I used regular nonpolar electrolytics because they were cheap and easy to find. The ideal is metal film because they're precise and last indefinitely. Even my crummy replacement capacitors made voices clearer and music better.