I believe that the actual answer is that nuclear weapons can be accessed and fired remotely, but only through authorized control channels. This is a fallback in case the chain of command is decapitated, or the chaos of a nuclear strike makes establishing communications difficult or impossible.
In the US, the authority to release nuclear weapons for use sits with the National Command Authority, which is the President (or duly appointed, legal successor) plus the SecDef (or their duly appointed, legal successor).
However, they have the power to delegate launch authority to someone else. That is, the NCA does not have to immediately declare a launch. Instead, they can give a copy of the Authorization Codes and Weapon Enablement codes to another party, such as the a high-ranking general officer aboard one of the various flying command planes. In this case, they've effectively legally devolved the decision to someone else; this usually comes with instructions on what conditions the launch is to happen. Legally speaking, that person now has the right to declare a launch. This capability is retained until the NCA decides to revoke it, and take back control itself.
So, ultimate, the authority to launch sits with the NCA, but it can decide to delegate this authority to others, after the NCA has agreed that a launch is authorized. So, the actual launch order might come from someone other than the NCA itself, but the codes will always be the same, from the perspective of the people who launch the actual weapons. While this isn't quite the same as a general on board an aircraft launching the missiles himself, it puts the line of control and responsibility much closer to the weapons.
In the former Soviet Union (and so far as anyone knows in current Russia), an automatic system known as "Perimeter" (or colloquially "Dead Hand"). This is a remote launching system with the ability to automatically launch the Russian arsenal if there is no communications with the centre command authority, and if the sensor system detects the signs of a nuclear attack in Russian Territory.
It is not known if China has any systems in place analogous to Perimeter, or the elaborate chain of command in the United States. Other nations are thought to use an authentication system, permissive action links and multiple crew working simultaneously to fire nuclear weapons, but no one knows for sure (for obvious reasons).