5S should do the job. Apple’s iPhone product cycle is annual with a new device coming out in September each year*. It’s new model in year one, then S model the following year. As they release the new model or the S, so the phones at the bottom of the range drop off – basically everything moves down one.
The 5S is still sold, but is their bottom of the range phone now. The assumption will be that it will be removed from the range in September. It will, however, be supported for a while longer.
As to battery issues, it’s possible to pay for a replacement (legit Apple or cheaper non-legit).
*they don’t ever formally say that’s true, even within the much of their own organisation. Paranoid doesn;t come close to describing how they operate!