Cobweb cleaning is one of my most hated barn chores at my place. If I were paying full board, I would expect that would be included.
That is, if they had a concern and wanted them cleaned up.
Having said that, when I did board I was very concientious about picking up after my horse, sweeping the aisle when I was done etc. But to me that is quite different from cobwebs. Come to think of it, I don't remember if they ever did keep them cleaned up and I don't remember thinking anything of it.
In my own barn, I hate 'em with a passion, hate cleaning 'em with a passion but it makes me feel better when it's done.