The ability to televise certain executions exists because some jurisdictions provide for surviving victims and their families to observe the execution if they wish, and there are cases where some legally recognised "interested parties" cannot attend in person. But access is targeted and restricted. I suspect that if legally challenged the Pentagon will try to argue that because of the nature of the crime the US general public are victims of this man's actions and therefore should be able to observe his execution. Federal courts have rejected "public interest" arguments in the past but those have been attempts by the media or internet companies to broadcast, not government decisions.
Surprise that this has not been announced as a Pay Per View event with the proceeds going directly to Hegseth or other members of the Trump Administration is pretty widespread in the USA, but so is the cautious hope that the publicity around this situation as well as that of Christa Pike will raise public interest and awareness about the negatives of allowing the Death Penalty at all. Approval of the Death Penalty nationwide is roughly 53% according to recent polling.