How can I overcome academic "researcher's block"?
Forgive me but I'm going to just wildly speculate here and hope that you find a bit of value in it. Perhaps you are just the sort of thinker that thrives on the synergy of working with others and getting ideas bounced around and reinforced (or shot down). That doesn't make you less of a mathematician, of course, but just represents one way of getting all of the pieces together.
It seems like you thought of yourself up to now as the 'junior' member of your circle and now the others have died or retired and so you are left 'alone'. In fact, IMO, you are left, now, as the senior member, but you don't need to continue to be the 'only' member.
Try to build up the circle again with new junior members with whom you can interact, building up their ideas as well as your own. It will give you that synergy again, if this is indeed the issue (like I said, speculative). A local circle is best, of course, but today's communication resources give additional options if learn to use them intensively.
And you have the opportunity to become the master that those younger scholars look up to and revere as they grow into the profession. Very cool.
One additional factor might be at play. When a person tries too hard for too long at intellectual activity, the brain sometimes rebels and shuts down. A break is called for, especially a break that gets the blood flowing, such as physical exercise. Physical exercise that also engages the brain but in a different way from the normal research mode can be helpful. I use Tai Chi for this as it is a mind-body fusion exercise, though not aerobic. When younger, I used cycling and x-c-skiing and such though they didn't have the same mental component. But an hour or so break for physical exercise can sometimes invigorate you.
And since the mind works subconsciously as well as when forced, sometimes insights magically appear during or immediately after a break. Even a sleep break.
I am an academic with tenure. I experienced difficulties with research at one point. I had been working on mathematics for about 30 years. I share below what helped to tackle it. I am back in a creative zone, happily producing results--in fact delivering some of my best results.
Others have given great advice, including finding new research partners to work with. Can you take on a grad student? I always found conversations with grad students very stimulating.
Panic attacks can be addressed via breathing techniques (very helpful and simple to apply). There could be other causes for stress or blank page syndrome too (e.g. conditions affecting sleep such as apnea etc), so perhaps check your health with a GP.
Others suggested engaging in sports. I find swimming energizing and super useful prior to a research session.
A creative activity (other than maths), such as sculpting or writing (or drumming or portrait drawing as Feynman did) will help to regain creativity elsewhere, which you can bring back to research in due time. It worked for me.
Gain distance. If one is too close to the work and attaches too much importance to it, then the work can suffer. I work with far more detachment now on a topic that I find more important than any other I worked on before. I keep perspective, humor and balance in the work. I found better pacing than when I was younger. If you change your life style and get more distance from your work by socializing, sports or creative undertakings, then the distance and added perspective this brings will help to overcome stress. I never stress in research at this stage. It is delightful (again and far more so than when I was a younger scientist).
Get a sabbatical to work with a mentor. Finding the right mentor to work with can make all the difference. I found a mentor who is very active, yet retired for quite a while. He is an inspiration and working with him and seeing him in action has changed my work practice.
All of the above helped me to move forward. Life always brings many challenges. A research block can be a sign that you require something different. So perhaps consider it as the opportunity that it is to regain what you are looking for...
Note: after the OP updates the question, I feel this answer doesn't answer it much. Nevertheless, it's still applicable in a general case I think.
The method to solve mental block is very easy: take a break. Everyone has it, even your brilliant collaborators, so there is no need to be anxious about it. (The more you are anxious, the more likely you have it).
My experience is that if you don't know what to write, read more books/papers. During the reading if you catch relevant idea, then it will invoke the idea that you are looking for, and then it will come as flood. If it still doesn't come, then perhaps ideas for other problems will come?
Other resources:
- Writer's block
- Occupational burnout (for a psychology approach on this)
- Process theory of composition (for those who is interest in the field studying about writing: composition)