The relative wind speed produces, as you say, the same force. That force, multiplied by the speed of the bicycle, is the power required. The power required is not the same in each case.
Sorry to say but that is complete nonsense. Not quite sure where do you get this sort of intuition.
A bicycle as far as I know without the brakes applied will roll backwards (unless you have some sort of special bicycle that will not allow that).
So just sitting still not pedaling you will act like a 0.5m^2 sail (may be less depending on your position on the bike but we will use this round number).
A sail of that size in 35km/h = 9.72m/s will provide an equivalent power of 0.5 * 1.2 * 0.5 * 9.72^3 = 275.5W
So if you want to drive upwind at 1km/h in a 35km/h headwind you will need to provide 0.5 * 1.2 * 0.5 * 10^3 = 300W
Bicycles are fairly efficient so if you ignore the rolling resistance and small frictions between pedals and wheels you need to provide 300W to maintain 1km/h with a constant 35km/h headwind.
Have you ever used a bicycle or pedaled upwind ? It is the same principle as driving on an incline. Maybe it is not that windy where you live.
How about ever being in a river and trying to move upstreem vs downstreem. Curious if any of you ever left the house
