Bennett Sorbo had an iPhone 6s with a dying battery. He ran his iPhone through some benchmarks, timing and filming the whole thing.
He then went to Apple, got a replacement battery, and ran the same tests again. The video below shows the results.
Bottom line, replacing the battery clearly speeds things up. If you jump to about 2:34 in, you’ll see that the tasks took 5:45 on the bad battery (a presumably throttled processor) and only 4:33 on the new battery.
That’s a savings of 1:12, or about 21%. Not necessarily accurate to say that your phone will be 21% faster with a new battery, but it certainly seems like a new battery would make your phone at least somewhat more nimble.
Good experiment.
If you look at the entire video, a big chunk of the difference was running Geek Bench. This will be stressing the CPU and memory and they will all be running at peek performance, this will increase the disparity between the two tests. At the point Geek Bench was launched the figures were 1:44 vs 1:27 this makes the difference more like 16% which is still a considerable improvement, but somewhat less than the 21% value quoted in the story.