I have found that FTP for me varies quite a bit based upon the terrain or technology chosen. Flat and slight downhill is harder for me to generate power than 6-8% grade uphill climbing. I generate the best power climbing in that sweet spot percent grade on my road bike on the hoods.
I suck on the trainer and generally derate my workouts 7-10% to get the same results as outside.
Fixed gear on the indoor track is different again and riding my time trial bike is different from riding my road bike. My time trial bike is harder to generate power.
Stava uses calculations and makes assumptions about CdA and rolling resistance and wind can vary a lot. How could they get that correct for all users and conditions. How would they know if you were riding on the hoods or in the drops or riding invisible aerobars? Since climbing is generally slower speeds and done on the tops or hoods, the effect of CdA is reduced. So I suspect climbing power results on Strava are okay with “normal” wind conditions. Everything else is up for grabs.
Assuming the trainer is calibrated and working, it is still going to be off from what you can do on the road but may be indicative with a fudge factor.
Hence, I ride trainer, road, track and time trial and have an idea of FTP on each and have a fudge factor to apply. Then, during the workout, I may adjust the workout to raise or lower FTP due to my training and which equipment I have been riding. If I have been riding my road bike on the trainer, I know that my outside power on the same bike is going to be different - up or down.