As someone who races hill climbs I wouldn’t use strava. I’d go as far to say I would not enter a hill climb race that used strava as its mechanism for timings. I’ve seen some very different results on Strava/Garmin compared to official times during races. Variability in GPS accuracy, cover, recording mode etc. can make things go wrong.
If you do decide to do it I would suggest you set the strava segment to be shorter than the course. Most errors I’ve seen have been when you are waiting very close to the start line or don’t ride very far past the finish line and it still thinks you are on the segment.
Also make your course long. A couple of seconds on a 2 minute climb will drop you right down the placings where as on a 15minute effort it makes less difference.
Don’t be fooled into thinking this is just a fun friendly event either. If it is a competition people will care about there position.