Occasionally, trees, shrubs, or stumps will spawn randomly on my map. If I have a worker free, I can spend some elixir and hope to find some gems in them once they are removed.
Is there any benefit to not removing them?
I do see workers (not builders) go to them and gather stuff from the trees / shrubs / bushes. Is that just an animation or does that help my base in any form?
This wiki contains all you need to know about trees, shrubs, bushes, and rocks, also known as 'obstacles'.
There is one small benefit to keeping them, an attacker cannot spawn troops on top of an obstacle. However, this benefit is trivial at best.
There are much greater benefits for removing them, gems and experience! You can get a small amount of gems and experience for removing and continuing to remove these obstacles. There is even the 'Nice and Tidy' achievement which will give you additional benefits after clearing 5, 50, and 5000 obstacles.
And you never see benefits from the workers gathering stuff from the vegetation. Theoretically they are feeding themselves. But gems are much more important than fed workers!
By leaving trees, scrubs and other kinds of obstacles in the way you expand a no spawn zone near your base that thus precludes enemies from spawning close to your outer walls or battlements. This allows you to have more time to fire upon any approaching enemies with your point defenses thus wearing them down before they reach the walls and potentially breach said walls thus giving you an advantage that's significantly better than in the mere benefit of getting gems or experience points.
