Yes, when I got the Macbook battery needs servicing thing couple of years ago, the processor was permanently in a low power state of 800Mhz. You can verify it using the Intel Power Gadget thing or terminal.
I have a 2011 Macbook Air (13-inch) and my battery has a “Service Battery” alert. It’s very slow when the battery is not at full capacity but it’s fine when the battery is full.
I ordered a replacement battery off Amazon last week.
Be aware that most "genuine" replacement batteries are counterfeit. I bought a replacement battery for my 2011 MBA too, but it only worked good for a couple of months.
