Just spent it a new battery very easily for 7 € for the years coming.
However do be aware that your perception of what is slow changes as you use newer devices which perform better, making older ones feel slower than they did before - much like a shiny new work PC makes your home PC feel slow (or vice-versa) - so that may partly explain it.
Finally, I've just breathed new life into my ageing Nexus 7 (2013) tablet by installing the custom ROM "AICP" on it, which has made a huge difference to its performance & battery life. I recommend giving it a whirl on your SO's S3 - download from http://dwnld.aicp-rom.com/?device=i9300 (though you'll need to read up elsewhere about installing custom ROMs if you've never done it before)
Point is, I won't change my S3 until it crumbles to dust, if I can help it. I don't understand the obsession with buying the latest mobile every 1-2 years.
Also, the GC on older Android devices is a mess. It does not take many apps hanging out in the background before things slow to a crawl.
Is it still getting security patches? If not then it's not running perfectly fine.
2 years update + 1 security for mobiles that cost more than a computer is not something I am willing to pay a premium for.
Do you have an study available that I can refer to?
My S3 is 4 years old now, and it is works perfectly fine.
When it dies, I will most likely adopt one of my Lumia devices as main one, or will buy a 2nd hand Android device, instead of giving money to support bad OEMs