Is it possible, that old smartphone becomes slow simply because it is old?
Performance has nothing to do with battery voltage. The reason is that the flash memory is "worn" and therefore slower and less responsive. 
My Nexus 7 (1st generation) is also very slow now and I have almost no apps on it. You can read more here: Why Android Phones Slow Down Over Time, part "No Solid-State Drive TRIM"


  No Solid-State Drive TRIM
  
  The lack of proper TRIM support was the main thing that caused
  Google’s original Nexus 7 tablet to slow down over time. This was
  fixed in Android 4.3, which added proper TRIM support. On Nexus
  devices, updating to Android 4.3 will fix this problem.
  
  If you have an older device that doesn’t have Android 4.3 and has
  slowed down over time, you can perform TRIM by rooting it and using
  the LagFix app. This app runs the same fstrim command Android 4.3 runs
  in the background. TRIM is necessary because of how solid-state drives
  work — solid-state drives slow down over time because flash memory
  cells must be cleared before they can be written to again. TRIM
  preemptively clears cells that contain data from deleted files,
  ensuring things will be as fast as possible when Android needs to
  write to those cells in the future.
  
  How to Fix It: Root your device and run LagFix if you’re using an
  older device. This happens automatically on devices running Android
  version 4.3 and newer.