The BA role can be stressful because it's a really important one. BA's are agents of change. Change isn't always by choice, and is never easy in any case. To address your points:

1) True. This is because there can be many challenges in a business, arising from many different points, and only BA's have the perspective to address those challenges. But this also has some positive benefits. First, it allows a BA to quickly become indispensable by picking up skills on the fly as needed across an organization. Second, lack of definition gives the BA the opportunity to create that definition themselves. If a BA has an interest in analytics or process modeling, for example, it's easier to find your way to that kind of work when you are not strictly placed in a role.

2) I personally haven't found this to be true. This may happen in an organization that is IT-centric or that doesn't understand the value that BA's bring.

3) Every organization has politics, and everyone in the organization must learn to navigate them. The successful BA is one that can provide leadership within the context of the politics and culture that exist. 

4) If not the BA, then who? Managers and users have very specific points of view that may not account for business needs as a whole. While it may be a stretch to ask a newly hired BA to represent the needs of a business, an experienced BA should be prepared for the challenges of that role.