A little buddhism goes a long way here, it seems.
There is a slight difference between being stuck and being obsessed.
As the saying goes, doing the same thing over and over again and expecting a different result is the definition of insanity.
So, if by stuck you mean obsessively trying to remove a single obstacle despite it's not being ready to move, I'm not so sure of the value of this.
On the other hand, if by stuck you mean that you are filling in more and more detail around the obstacle in order to "soften it up", then this is quite a healthy process.
I mention the buddhism, because "it is stuck" is much healthier than "I am stuck" (although a buddhist may not even acknowledge that anything is stuck at all). Mathematicians are a bit too obsessed with the "I", and this creates undue emotional distress and paralysis. As you mention, this situation is much worse for those who don't have the luxury of time. This creates a very sad situation, because being all bunched up like this really messes with the ability to relax and perceive what is right in front of you. Rather than naturally growing a point of view, you end up endlessly beating yourself up for trying to repeatedly stuff a square block in a round hole.
If you stop taking things so seriously, you will find that it is enough to attend to the problem and obstacle fluidly, and enjoy the process of exploring around the obstacle. A good problem provides many opportunities to do this.
You might ask what constitutes a good problem, even. Stubbornly chasing the accolades of breaking a hard problem looks a little silly if you stand back far enough, especially when contrasted with enjoying the unfolding of a breathtaking structure that is "ripe", so to speak.
I get the impression that too many people are in love with the idea of being a mathematician, more than with the mathematics itself. If you honestly investigate the quality of your experience in doing math, maybe a break is in order, or a career shift, or a problem shift. Good mathematics is supposed to be fun. I think we forget this much too often!
Regarding the career trajectory, there are economic realities to consider. Everyone feels that their work/research/contribution should be valuable to the development of their subject. This is certainly true, but I think we tend to overvalue our own contributions. In the grand scheme of things, only a few of us will find a fruitful enough direction to support a career in research. (I am not one of these, although I've had some rewarding ideas.) If you put the "I" on the back burner, this is easier to accept. Maybe, despite your interest in obsessing over a problem, your effort would be better spent elsewhere? Maybe you fit into the world differently than you plan to? Ironically, accepting this reality may be the best way to get "unstuck" on your problem.
Once I was talking with Peter Jones from Yale about this sort of thing. I was talking to him about how hard I was working on such and such and how I had to learn to do this and that thing better and change my focus in such a way. Peter said to me, "That's funny, I always just did what I liked!" I realized afterward that there was a sobering reality to this. He just fit into his part of the mathematical world a bit more naturally than the rest of us. All of his experience perhaps led naturally to where he was. Others, like me, were forcing it.
Here is a Zen Koan:
A monk asked Tozan, "How can we escape the cold and the heat?" Tozan replied, "Why not go where there is no cold and heat?" "Is there such a place?", the monk asked. Tozan commented, "When cold, be thoroughly cold; when hot be hot through and through."
So there is an answer the OP's question akin to the above Koan:
When stuck, be thoroughly stuck.