The flights climbed number for me is off. I'm not surprised at that, FitBit got rid of the feature for the same reason (it's really hard to get right), but I'm a bit surprised that since it's not very accurate it was included at all.
When I experimented with my FitBit I had to exaggerate my vertical motion when going up and down stars to get it to register well. It seems I'm too 'smooth' walking up and down stairs to reliably trigger the mechanism.
Sure it is. Users are already getting inaccurate 'floors' measurements and posting the associated bug reports in their forums. If Apple had paid attention to FitBit's featureset over time Apple would have realized the feature was removed because it was uselessly inaccurate. A barometric sensor doesn't become more accurate for tracking flights of stairs climbed just because it's in a device with an Apple logo.
My Fitbit reports the number of flights climbed - it's the Fitbit 'One' model. Along with my daily 10k steps, I have a flights climbed goal that I track through the website. Maybe it's a different model that doesn't track this?