The Nexus 5 is 8 years old now. (The age is a particular problem given its lack of replaceable battery.)
You can find used budget smartphones from 3 - 4 years ago for about $20 and they will have far more performance and capability.
From a security and privacy standpoint, you should probably run a recent operating system. That's a tricky one with smartphones, Android is rarely maintained properly by the hardware manufacturer for long. A device supported by something like LineageOS or similar would be good to look for.
As a starting point, the budget Motorola series (Moto E, Moto G series) tend to have small screens, large batteries, and unlocked (or unlockable) bootloaders, often with maintained LineageOS ports.