"Why are theoretical physicists/mathematicians not rushing into this field? If it was so universal and apparently simpler to learn (also as claimed), then why is it not as widespread?"

Theoretical physicists already have mathematical techniques that work. GA may be simpler to learn, but that is no advantage to them. They have already learned the trad techniques. The easy thing for them to do is to keep using them. There is no cost to this. Learning GA would be a cost.

According to Thomas Kuhn, nothing new will come into a field until there is something the new technique can do that the old one can't. GA doesn't qualify. It can give you no answers that the old ways don't. Simpler, easier answers in some cases, but not to the degree that would attract great enthusiam.

OK, suppose a student learned GA first. He/she would still need to learn all the trad techniques in order to have access to the vast body of literature. MORE would have to be learned: all the trad techniques plus GA. Would this approach be an advantage? As far as I know such a student does not yet exist so we can't say.

What I can share is my own personal experience. My unusual interest is what physics would be like if we had more spatial dimensions. Much of the trad math does not work at all there and can't be adapted. In particular, applications of the cross product work only in 3D. I had to use geometric algebra instead.

I haven't gotten that far, but already I can tell you that I found this quite valuable in having given me insight into trad 3D magnetism. Describing magnetism in terms of planes of rotation is natural. That's what magnetism is: a force that causes curves and rotations. Describing magnetism in terms of vectors produced by a cross product is unnatural. Sure, it works. But having experienced both, I can tell you that the trad approach gives the wrong intuition. Going further, quantum spin is notoriously unintuitive in 3D. It only makes sense as a rotation or cycle in even-dimensional space, and GA handles that easily. Maxwell's equations make more sense in GA. The signs come out in a sensible way, which they don't in the trad method. I'm not sure why, but suspect it is because GA doesn't have the arbitrary right hand rule.

Now a thing about any field of study is this. Things are taught a certain way. The field then promotes students who easily learn in this way, while rejecting students unable to learn in this way. Perhaps those students could learn in some other way, but we shall never know. Those accepted students eventually become teachers, and teach in the way that they found easy to learn. So the system quite naturally perpetuates itself. If some new way of learning comes along, well, teachers are very busy. Do they want to become beginners once again? Usually not. Even if they would like to do this, the time is not there. So the system naturally resists change. It will continue to do so until, as Kuhn explained, something comes along that is so much better than the old system that the considerable effort to change is seen as worthwhile.

Now a bit about my personal experiences. I learned in math grad school (rather to my surprise) that I was no good at math. I had done well in lower math but that was by unconsciously using homebrew geometrical methods. That can be very effective on an SAT test but not in higher math. I couldn't do it. All I could do was geometry. That's how I think. So GA is perfect for me. I find it easy to imagine subspaces and so forth. My officemate couldn't do that at all. GA might have no attaction to him, nor to great majority of mathematicians suspect I. Perhaps they would find it impenetrably abstract. Like I noted before disciplines tend to select a certain type of person, and this perpetuates itself.

There is hope for GA though. Books that teach physics in GA are starting to come into existence. There is nothing to stop people like me from learning in this way.