Well... You can. Sleep deprivation is one hell of a drug. After ~23h on a plane without sleep, I slept through a fire alarm at a hotel. It's unlikely that both pilots and other staff would ignore it, but "no one can sleep to" threshold is not that low.

Have the alarm go off in the cabin as well.

That doesn't work at all. There are often times like takeoff, landing, dealing with adverse weather events, re-routing, mechanical failure, etc where they're going through a detailed checklist. The last thing you want is some random device that takes attention away from them dealing with a serious problem.

That's not impossible, but the pilots' limbs are already largely "spoken for" by other controls and tasks, so it would be a harsh tradeoff.

If other controls are being actively engaged by the pilot, the need for the button press would be delayed only until there's some continuous period of non-engagement.

Exactly - something like positive activation every minute of some control or another or it starts hooting.

Sorry but this a bit ill-informed.

First off, you cannot just put an alarm on everything. The sonic experience inside a cockpit is very carefully designed to give pilots the correct information at the correct time. False alarms are not only not appreciated, they are actively dangerous. If a flight experiencing an emergency situation, a blaring alarm that is incorrectly going off can prevent pilots from getting timely information.

Airbus have side-sticks (like a game controller), not yokes and you really shouldn’t be inputting on them unless you’re hand flying for whatever reason.

In all two-pilot aircraft (all commercial jet aircraft), there are two separate roles that each one assumes: Pilot Monitoring (PM) and Pilot Flying (PF). The PM’s job is usually to run checklists, communicate/operate the radio, check various indicators, and support the PF to fly the aircraft. The captain and first officer swap between these roles en route. The PM should NOT be inputting controls unless there is a good reason to do so, but for safety/redundancy, one side’s controls are never disabled unless a lockout/override is active.

Most airlines have a policy to cruise with the autopilot on, which keeps the aircraft on its plotted course at it’s cleared altitude. The time where “flying” comes into play most often is during takeoff and using various levels of ILS from just indications of glide slope to a full autoland.

While not fantastic, a cruising airbus will keep its current course and stay airborne if the pilots snooze off.

Charlie Victor Romeo was a very interesting watch for that cockpit fly-on-a-wall experience

A proper deadman's switch can't just be a button - you could weight it down, or just fall asleep on top of it. Trains commonly use a pedal you have to hold down halfway. Not sure this is really the best idea though, as pilots typically need both hands (and usually feet) free to actually, you know, control the plane.

A button press could be required, though.

If the pilots are controlling the plane, they are doing detectable actions and pre-empting the dead person switch.