Why won't my camera fire the off-camera flashes even when the trigger button will?

You need all triggers on TRX. Period.

Here is the break down for what will happen with different trigger settings:

Trigger on TX: pressing the button fires all flashes on a trigger with TRX setting, as well as any flash mounted on the trigger itself. It does not trigger any camera, not even a camera connected to the camera trigger output. The hotfoot contacts will have the same effect. You can use a trigger on TX on a camera that gets its shutter signal from somewhere else. When pressing a button on such a trigger, it is a flash setup test button but triggers no camera.

Trigger on TRX: pressing the button will trigger all cameras connected to the camera output (2.5mm TRS) of a trigger with TRX setting. It will not trigger any flashes. When the hotfoot gets triggered, however, it will trigger all flashes on a trigger with TRX setting, including the trigger with triggered hotfoot itself.

So basically the rule is: if you are remote triggering a camera, all triggers need to be on TRX setting. Including the one you are triggering the camera from. If you put it on TX, you can use it for triggering all flashes without involving the camera. For test purposes, or because you put it on a different camera and want to take a picture with your flash setup on that camera (using timer or just its normal shutter button) without triggering the other camera.

A trigger on TX will never trigger a camera, but only flashes.

Counterintuitive to some degree, but very useful.

EDIT: after rereading it would appear that you are not actually triggering the camera remotely and your switches are set up correctly.

The answer for that case is: check that the camera has flash enabled and set to "external" and that the flash trigger is fully pushed onto the camera hotshoe. The pins on the Yongnuo trigger have comparatively strong springs and comparatively shallowly rounded contacts. It is very easy for the trigger to feel stuck before it has indeed been pushed fully on.

In particular, the locking sleeve should be fully up before pushing on in order to have the locking pin fully retracted.

Here with locking pin retracted:

Here with locking pin out: