A 5S will still run the latest iOS, and with a new battery wont be under “performance management” mode, so it should be quite fast to use.
I just checked and there are a couple hundred listed on my local Craigslist for $100-$150.
Are you suggesting some official refurbishment route or just find them on Craigslist or Ebay?
It was night and day in terms of performance. In particular, Siri works much better.
Developers assume the performance of the latest, so they don’t realize how slow things act on older devices.
I don't know what it's like in the US, but here (SE Asia) there is a massive market for second-hand smart phones/tablets, usually from places that also do repairs of such devices. Go to any shopping mall in Thailand and (usually on the top floor) there will be dozens of places with devices of varying age.
I don't think Apple's official refurbished stuff will be in the price range you're looking (they only have 7's from what I can see) but I'm sure you can find them somewhere.