Why don't aeroplanes automatically put out engine fires?
Shutting off the computer is difficult from a technical implementation standpoint. You're handing over a lot of power to the computers and sensors letting it shut down an engine, where these usually do not have access and control. This is certainly not always a good idea.

Pulling the fire handle is a pretty aggressive move: It will typically disable hydraulics, electrical power, and bleed air coming from that engine. If you're not prepared for it, it can end badly.

The engine will often be partially functional, depending on the type of failure. If you get dual engine fire indications, you'd probably wait with shutting them down, if doing that at all. This sort of behaviour would have to be coded into the flight computers.

I don't think there are any catastrophic consequences of not switching the engine within 8 seconds rather than maybe 2 seconds. It does not make a great difference. To set off the detectors, it must already have spread a little. I don't think you could do it much faster with the pilot staying updated on what's going on.

Needless to say, computers sometimes get things wrong. There is a human level to check what's going on before proceeding.

There are times when you don't want the computer doing it's own thing. Airbus states that:

If an engine fire occurs during takeoff or go-around, the Pilot Flying shall first establish and stabilise the aircraft on a safe climb path and then proceed with the ENG FIRE procedure.



Footnote:
It could certainly be done, but with the pilot-computer relationship on current flightdecks it's not a good idea.