Trackpad left-click is unresponsive - MBA Tap-to-click works fine, and so does right-click - most of the time. Left click does not work at all, unless you press it very hard at the very corner and hold it there for a while until it registers, although it worked a little sporadically yesterday, worked completely normal the day before.

The computer was manufactured mid-2013, but the battery was in 2016 and replaced mid-2017 - that's all the modifications done to it.

I'm guessing it's a hardware problem, but the clicking sound and feel is completely normal.

Any ideas about what is at fault and how to fix it?
The 2013 MBP has still the mechanical trackpad. Check if the trackpad is stressed physically in some way. When the battery swells it happens that the trackpad stops registering clicks, because it can not be pressed.