With that setup you need a central control that the buttons toggle on or off. I imagine that would be more complex than just running a cable from one switch to the next. Not exactly the most user friendly solution, however possibly the cheapest.

Not really. I was imagining a toggle switch button—like the kind that goes in and out to express its state, where the “in” state is literally bridging contacts with the back of the button—but without the in/out state being exposed externally, since it’s not accurate to the state of the system as a whole. Instead, it would be a spring-loaded toggle where the spring is less compressed when on and more compressed when off, to make the button face remain at the same push-depth either way, while still having an in-out switch behind it. Like the mechanical NumLock key on early keyboards.

So each switch-button is still connected in series; all the LED indicator faces on the switches are powered by line voltage of the circuit; and so, when the circuit is completed, all the switches become lit; and when any switch is toggled, the circuit is broken, and so no LED is then receiving power.

You can have a mechanical button toggle the position of the interior switching hardware. Every press will flip the switch inside to the opposite path. Like the button on top of a retractable pen.

Maybe a bit more complicated; there are moving parts required to index the action. But doable without extra wiring or centralized control.