You need to consider the very last sentence you wrote.
There are 2 factors.
Angle and Rate.
I can just see myself explaining this so badly.....................at this time of night.
Aerodynamically, the aeroplane does not care what direction the wind is travelling. It is only concerned with its speed through the air.
Lets assume 2 numbers for the story...75 knots = 500fpm climb
So if you are climbing at 75 knots in still air, the aeroplane only see's 75 knots of airflow.
If you are climbing at 75 knots with ten knots of headwind, then the aeroplane see's 75 knots of airflow.
If you are climbing at 75 knots with ten knots of tailwind, the aeroplane still see's 75 knots.
For rate of climb, you do not need to consider the direction of speed of the wind. The aeroplane will travel through that pocket of air and generate the performance you wish.
This is RATE of CLIMB, or descent.
Where the wind comes in, is how it affects your distance over the ground....or ANGLE of CLIMB.
IF you have a headwind, and you fly the aeroplane at 75 knots climb, it will cover less ground distance than nil wind, so the angle is steeper....but you are still climbing at 500fpm....from sea level that is 4 minutes to 2000 foot....and without a calculator...........say 5 NM for nil wind, and 4nm for 15 knots head wind.
If you have a tailwind, it remains at 4 minutes to 2000, but now it is going to take 7nm.
Summary.
Rate= performance through the air
Angle = performance over the ground