Full stack provides something that silo'd positions don't: variety. I am a full stack developer. My title may be Director of IS, but that's just a title. I'm the primary designer, architect and implementer of every feature in our system. Back end, front end, middle tier. You name it, my hands are on it. The problem is, I'm the only developer in our company who can do that. We have two other developers, but their skills are narrow, and they're just as busy as me in their respective areas. So, while it provides lots of variety, it also means I'm the person everyone runs to when things aren't working right. Sadly, the company cannot quite afford to hire another developer who is more well-rounded. (Full stack developers tend to be significantly more expensive.)
As someone in a management position, I can tell you this: I'd rather have one full-stack developer on my team than 5 silo'd developers.
Now, you mention burnout. Yes, that is very real, especially if you're the go-to person all the time. I have not figured out how to avoid this. It's completely cyclic in nature with no break in between. Here's an example of the cyclic routine of a full-stack developer:
No break. No time for vacation. Repeat ad nauseum. This is the life of a solo full-stack developer. When you are a solo full-stack developer, often, things really do fall on your shoulders. It gets really old after a while.
I really like working for small companies because it give the opportunity to be all over the code base, and even into hardware stuff. And, full-stack has its rewards. But being the ONLY full-stack developer in a small company can make one burn out very quickly.
I definitely would not say steer clear. Instead, I would say, set ground rules with your employer. They give you paid vacation? Take it. And tell them you're refusing to get caught in the cycle I described above. I need to take my own advice here. I'm terrible at this. But if you want to be full-stack AND avoid burnout, you need breaks to regroup.