I stumbled on an accidental fix. Try this: let your Apple Watch battery die. After that, wait 3 or more hours before charging it again and see if that resolves the Flights Climbed over-counting issue.
Below is the full story:
For about two months my AW4 was way over-counting Flights Climbed. Yesterday morning I noticed that Flights Climbed was 0 after I made my breakfast instead of the usual 10 or so. I have been monitoring since and I can confirm that the issue has been resolved. Story below and I would be interested to hear how it started (and hopefully ended) for you as well.
On April 4th I was working on a large yard project and decided to run continuous heart monitoring through the Cardiogram app. I ran it for 10-12 hours. Flight over-counting started precisely on that day, over 200 flights climbed (in reality it was probably less than 10). Watch continued over-counting till May 31st.
On the morning of May 31st, I decided to do a 45 mile bike ride, which is more than any exercise activity I have recorded on the watch previously. The watch activity app ran for 4 hours, 3 of which I was active. Watch continued to over-count flights that day.
May 31st night, I decided to sleep with my watch on without realizing that the battery was critically low (due to the 4 hour activity). Watch died around 2:15AM, or at least that was when the last heart rate reading was taken. I put it to charge when I woke up at 5:00AM and left it till 8:00AM. Since then, over-counting has stopped. The only two special events that happened prior to this resolution were the extra long exercise and the battery death. I am leaning more to believe that the battery death is the culprit.