I would not say a trailer adds stability. The connection between the bike and the trailer is somewhat flexible, so there's a constant yoink-yoink going on when you are going up even a moderate uphill. And the physics of pulling a trailer also mean that sometimes the trailer begins a small downhill when you're already on the flat. That means, you suddenly get accelerated as the trailer reaches the dip and rolls down it, pushing you forwards. Also when you need to stop fast, the physics are a bit different than when you are riding just a bike.

The difference is not huge, but if something, it makes holding the balance a little bit more difficult.