“feature branches” is when each feature is developed in its own branch and only merged into the main line when it has been tested and is ready to ship. This allows the product owner to choose the features that go into a given shipment and to “park” feature that are part written if more important work comes in (e.g. a customer phones up the MD to complain).
“refactoring” is transforming the code to improve its design so as to reduce to cost of change. Without doing this continually you tend to get uglier code bases which is more difficult to write tests for.
In real life there are always customers that have been sold new features and due to politics all the customers have to see that progress is being made on “their” group of features. So it is very rarely that there is a time without a lot of half-finished features sitting on branches.
If any refactoring has been done, the merging in the “feature branches” become a lot harder if not impossible.
Do we just have to give up on being able to do any refactoring?
See also "How do you handle the tension between refactoring and the need for merging?"
My view these days is that due to the political reasons that resulted in these long living branches and the disempowerment of the development director that prevented him from taking action, I should have quicker started looking for a new job.
Feature branches certainly make refactoring much harder. They also make things like continuous integration and deployment harder, because you are ballooning the number of parallel development streams that need to be built an tested. You are also obviating the central tenet of "continuous integration" -- that everyone is working on the same codebase and "continuously" integrating their changes with the rest of the team's changes. Typically, when feature branches are in use, the feature branch isn't continuously built or tested, so the first time the "feature branch" code gets run through the production build/test/deploy process is when it is "done" and merged into the trunk. This can introduce a whole host of problems at a late and critical stage of your development process.
I hold the controversial opinion that you should avoid feature branches at (nearly) all costs. The cost of merging is very high, and (perhaps more importantly) the opportunity cost of failing to "continuously integrate" into a shared code base is even higher.
In your scenario, are you sure you need a separate feature branch for each client's feature(s)? Could you instead develop those features in the trunk but leave them disabled until they are ready?. Generally, I think it is better to develop "features" this way -- check them in to trunk even if they aren't production-ready, but leave them out of the application until they are ready. This practice also encourages you to keep your components well-factored and shielded behind well-designed interfaces. The "feature branch" approach gives you the excuse to make sweeping changes across the code base to implement the new feature.