Drawing API does not have anti-aliasing when drawing to a texture, and therefore you can see such a ladder.
It cannot be fixed.
You have two ways to work around this:
1. Use Tileset. This mode works differently and will not have ladders.
2. Use some third-party Drawing API.

I am wondering what this jitter comes from and whether I can smooth it and how.

How does the route function return from Google API? Is it giving a discrete number of latitude/longitude pairs which this line is then the direction drawing of from one to the other?

Is there jitter because it is drawing only horizontal/vertical between the points (step ladder)? Or is there jitter from noise in Google API's results?

Could it be smoothed by interpolating or local averaging?

Which function would I want to manipulate to fix this problem if it is fixable?

Thanks.