Why are on-board computers allowed to change controls without notifying the pilots?
There is a general design principle, that some, but not all, of the behavior of the flight guidance system or autopilot should be visible to the pilot.  Usually automatic engagement or disengagement of a control system is indicated, but sub-modes of these controls or manual changes might be unindicated. That sounds simple and logical, but in reality it's a complex and tricky human factors issue. Displaying too much information to the pilots is just as bad as displaying too little, as too much information can cause them to ignore the most important indicators, like a stick shaker warning about a impending stall.

I know this isn't a very satisfying answer, but it's hard for anyone other than a human factors expert to answer a very general question like this with hard-and-fast rules.

For example, many autopilots will skip the aural warning about disengagement if the disengagement was directly caused by a pilot action. The pilot in this situation only gets a visual indication that AP has changed behavior. This sounds perfect, but on Aeroflot Flight 593 a pilot let his child into the cockpit, who then applied enough pressure on the yoke to disengage the automatic aileron control. There was no warning chime because the disengagement was considered intentional by the autopilot. This was unlike previous planes the pilot had flown, which contributed to the pilot's confusion when the plane began to turn and then go into a sharp bank.

The indicator light you mention for the 737 Max Lion Air crash is for "AoA disagreement", not MCAS engagment as you suggest.  While these kinds of AoA indicators are generally considered good ideas, it's still an area of debate how much they'd help confused pilots like in the Lion Air crash or the Air France 447 crash.

Similarly, the quality of indication was not the biggest cause to the accidents you've listed. For example, the Scandinavian Airlines Flight 751 could probably have returned to the airport successfully, but a compressor stall due to ice striking both engines at low altitudes is a bad situation no matter the auto-thrust behavior. Bad sensors and confused pilots can crash a plane with perfectly designed indicators.

Edit:  since this answer is getting a lot of attention, I think it's important to note that I don't really cover any special considerations for the edge of the flight envelope, auto-trim vs auto-disengagement vs changing sub-modes, or idiot-proofing of flight systems. I'd love to see this covered in more detail by people with experience.