In my experience, I never have found that cryptographers base their opinion of a cryptosystem on the properties of the underlying group. If its a braid group, abelian group, or finite field: that does not really matter. What matters is, as @Thomas notes, how hard do we think the problem is in a particular setting?
Cryptography in braid groups usually has security reductions to problems related to the simultaneous conjugacy search problem. This problem has definitely not been studied as closely as problems related to integer factorization or discrete logarithms; and it is likely less studied than finding multiplicands on elliptic curves or the shortest vector problem on ideal lattices.
The community warms up to an idea after "adequate" attention has been given to the underlying hard problem. What is "adequate" is a mix of time and attention. It also helps to have a company pushing and standardizing it (Certicome with ECC, NTRU with lattices). It can also help gather attention if there is some "killer app" other than efficiency (ECC was helped by pairings, lattices will likely be helped by fully homomorphic encryption).
Braid group cryptography has none of these. Its main selling point is efficiency, which is a tough sell. It used to be embedded systems and mobile devices, then smartcards, and now RFIDs and sensor networks. As technology gets better, small computational devices become more capable of implementing standard cryptography.
Further, protocols like AAG have had a number of attacks against them requiring further refinement of how parameters are chosen. This isn't necessarily devastating: it could be viewed as akin to moving to safe primes or away from certain curves, or it could be a sign of deeper problems.
To answer your questions directly... I am not sure what you mean by a neural exploratory article. I don't think many cryptographers consider AAG secure or broken; the jury is still out (and no sign of them coming back any time soon). I don't think the group theory it is based on has any role in people's opinion (other than how the group theory dictates the hardness of conjugacy search problems).