I'd do it, even with the risk of buying a crap battery, especially with Arch. I'm typing this on a 2014 Lenovo with Arch and XFCE. The main thing it has trouble with is the external 4K display which is a bit sluggish--I turned compositing off to make it a bit better. The battery health is about 80%, last I checked, so I haven't bothered to replace it. (Also, full disclosure, this is effectively my "desktop", so it stays plugged in all the time; I keep the battery charged to 50%.)

But I tend to run computers until they die die, getting a perverse kick out of maximizing the use of old hardware. This machine used to be a Windows machine until it became so sluggish as to be unusable. Putting Linux on it was like dropping the lead weights in a sprint.

You don't buy a genuine battery, they are indeed too old. There are third party manufacturers making new batteries for old thinkpads, kingsener and greencell are two. I have kingsener in my homelab X230(Arch) and T440p(NixOS/Silverblue) and am very happy, basically better than new(more recent battery tech).

Depending on price I might go for it

I did spend like $40 on a battery for a 10 year old laptop a few years ago so that I could keep using it for troubleshooting a remote network I setup up awhile ago because it's cheaper than a new laptop.

Personally I'd see if you can find a new compatible battery not necessarily an OEM battery though as being new it's probably going to last awhile.

As a Linux toy, I’d absolutely replace it. I have a Zenbook from 2011 that still works fine enough for desktop apps with Mint after a battery swap.

At minimum though, I’d open it up to make sure the battery isn’t a bloated spicy pillow waiting to blow.

I’ve got a couple of machines that I put new batteries into not long ago. If it’s solid hardware, throw Linux in there and give it new longevity.

just open the battery you have and replace the bad cells

Back when smartphones had easily replaceable batteries, like the LG V10 I bought for my mom, I bought her a new battery after 3 years and it gave it new life.

Now, 10 years is on a different time frame. Personally, I would gauge the price of a replacement battery against how much I paid for the laptop. 10% maybe? Completely arbitrary. $20 replacement battery for a $200 used laptop.