Why are my object's shadows so janky?

Janky shadows are the result of janky topology: Large N-gons with non planar faces.

At the time of rendering all objects get subdivided into triangles.

I cringe just looking at this horrible mess.

The takeaway is this: objects (yes all objects) should be created with some notion of the topology. Fixing things later is much harder and time consuming. It has been written many times in this site "topology is your object", is not something that you add at the end like icing on a cake.

So how to fix it? you need to re-topologize. There is no de-janker tool...

There are a lot of videos out in the Wild World Net detailing how to do it. Basically it comes down to creating a new object, and using the old one as reference to pin new faces, using proper quad topology.

If you want to try blender 2.9 there are new tools to make re-topology much easier.

https://www.youtube.com/watch?v=BEwEWKOH5ws

After looking at your uploaded file, I would say you don't have enough geometry to describe the object without having non-planar faces, and using the Split Non Planar Faces command will only produce more non-planar faces as it tries to remove the old ones. You can see this if you do the same command again, it still splits faces. What you need is more even geometry, instead of long, thin faces, or 'curvy-edged' faces where the vertices describing those points are not on a plane. First, try retopologising to get rid of those long thin triangles, then add edges across your faces to produce quads or tris that are planar. You can also experiment with the remeshing tools in the remesh section of the object data tab in the properties window. I'm curious how you ended up with that geometry, it looks you you used some non-standard workflow. Nothing wrong with a non-standard workflow, but sometimes, unless you know what you are doing, you can end up with problems like this.