I not specifically familiar with the proposed TN amendment, but the NRA-ILA did have a brief article on it prior to the election.
Hunting & Fishing is controlled by the individual states, and the ISSUANCE of a license is considered a priviledge in most states. And the state could potentially refuse to issue a license for a valid reason, though I doubt that they would do that, unless you were mentally incompetent or such thing.
Its my understanding that this amendment is to protect hunters & fisherpersons from harrassment, verbal abuse, and even violence directed at them from some extreme eco- groups such as PETA, Greenspeace etc. Laws can be changed, but a state constitutional amendment is a lot more difficult to change or remove once its on the books. It would be easier to prosecute a protestor in state court, than a local court, and would probably carry a stiffer jail sentence.
For example, KY F&W is self funded by license fees, and federal fees from the Robertson-Pittman act, no operating funds come from the General Fund. Any threat to these fees would put the KYF&W in jeopardy of managing the wildlife resources. Protestors could discourage and disrupt the lawfull use of these resources, and thus the income to the F&W.