I would recommend getting a new battery if you haven't replaced the battery. If you are looking for a logic board replacement,
# I would recommend getting a logic board with its paired home button. However, the only downside of getting the logic board without its paired home button is that Touch ID will not work.
# The color of the phone the logic board is from will dictate what color home button you get(black phones have a black home button, silver phones have a white home button with a silver circle, gold phones have a white home button with a gold circle). If you get a logic board without a home button, the color of the phone the logic board is from will not matter.
# As long as the logic board works, it should work with your phone.
# As more iOS versions are released, some apps will stop supporting the version of iOS on your iPhone 5s. If this iPhone 5s is not on iOS 12.5.4, you should update it to this iOS version as soon as possible.
# As I said before, I would recommend replacing the battery. As batteries are charged and discharged, they slowly lose battery capacity. Therefore, replacing the battery in an old and well-used phone can give it more capacity.