On iPhone 5s, can the battery itself be the cause of fast draining, even after a brand new U2 chip is installed?
Short answer

Yes, a battery's condition can most definitely explain why it is draining fast. iPhone batteries are consumable components that become less effective as they age. They have a limited lifespan and their capacity will decline over time, eventually leading to lower performance for your iPhone.

Long Answer

You will get lots of opinions on batteries, so when it comes to Apple products I always encourage users to read what Apple has to say on the topic.

Apple also recently published iPhone Battery and Performance.

To partially quote Apple:


  All rechargeable batteries are consumable components that become less
  effective as they chemically age. 
  
  As lithium-ion batteries chemically age, the amount of charge they can
  hold diminishes, resulting in shorter amounts of time before a device
  needs to be recharged. This can be referred to as the battery’s
  maximum capacity—the measure of battery capacity relative to when it
  was new.


I would encourage you to read the entire article.