In basketball, why win the opening jump ball if there's no advantage (possession will be rotated afterwards).
Because you want the extra possession in the 4th quarter.
The team that gains possession after the opening tap will put the ball into play at their opponent’s end line to begin the fourth period. The team losing the opening tap will put the ball into play at their opponent’s end line at the beginning of the second and third quarters.
Note for NCAA and lower - Jump ball is very important because all possessions are alternating including jump ball situations and second half possession. Original basketball rules settled things with a jump ball. Possession arrow was something to make things go faster and to be fairer to smaller teams (and to get rid of referees having issues throwing a good, fair ball for jumps).
No real reason; as you say possession will be rotated afterwards. Probably the biggest advantage is the psychological benefit of having the first possession and thus the first opportunity to take a lead, but I imagine that's a very minor benefit.