Taking your comments in turn.
"Now suppose we're on ground and we have a 20 knot tailwind and stationary.
The TASmeter and Airspeed Indicator would show zero as there is no -20 knot value"
The speed of the air flowing over the aircraft is now 20 knots from tail to nose. If the Airspeed Indicator and TASmeter were able to register this they would show -20 knots. They may not be able to show negative values, but this is what the airframe is actually experiencing. So as far as the airfame is concerned it is moving through the air at -20 knots (ie 20 knots backwards).
"the GROUNDSPEEDmeter would also read zero"
Yes this is true because the aircraft is not moving over the ground.
"and the anemometer on the ground would read 20 knots".
This would be true if it could sense only the speed of the air. But if it could also sense the direction, and it could indiacte negative values it would read minus 20 because the wind has reversed direction compared to our initial situation.
"If we accelerate to 180 knots relative to the ground, the TASmeter and Airspeed Indicator would read 160 knots"
This is correct. They would have gone from -20 knots to +160 knots reflecting an acceleration of 180 knots.
"and the GROUNDSPEEDmeter would read 180 knots.
This is also correct.
"The anemometer on the ground would remain unchanged as we suppose that the wind hasn't changed".
Yes it would still be reading -20 knots if it had been able to do so.
The important point to note is that in terms of lift and drag the aircraft responds to its speed relative to the air.
If we assume that lift-off speed is 170 knots, then with
20 knot headwind lift-off will be at 170 knot airspeed and 150 kt groundspeed.
20 knot tailwind lift-off will be at 170 knots airspeed and 190 kt groundspeed.
The real acceleration required is relative to the ground, so comparing the headwind and tailwind cases shows that we need more acceleration in a tailwind. This increases the length of the take-off run required to reach lift-off speed.