Yes, it is definitely possible to write flutter code that is way too deeply nested, just like in almost any programming language it is possible to write functions that are too big and complicated, or too deeply nested. Because Flutter views are defined programmatically, the solution for Flutter/Dart is the same as what we all know is the solution for any programming language: Refactor your code. For example, take out a subtree of widgets and make them into a new widget (object); hopefully a new widget that you can reuse in other places.
We also have tools that help you deal with large widget trees. Mixing styles with UI structure is a feature that makes it easier to create good and flexible layouts, but business logic should be separated from the view. We have been paying special attention lately to helping people structure their apps. You should look at https://github.com/brianegan/flutter_architecture_samples for some examples of good architectures.