I think most successful agile teams would adopt those non-functional requirements not as tasks to be implemented but as cross-cutting concerns of all work. Ie, on a Scrum team they could be part of the Definition of Done, ie "Is the code maintainable" which would not allow some work to be considered completed until it met that definition.

I'm not 100% clear on what you are saying but it sounds like you think agile teams ignore these non-functional requirements in favour of perceived progress on features. I think that may well be true in practice but it is also true of non-agile teams. It is in fact just true of immature development teams the world over. Being agile or not has nothing to do with it. Scrum and other processes do at least try to have something in place to cater for these requirements, which often sadly get overlooked when pressure is applied to make flat out progress on features.