Another way to look at this "independence of g" phenomenon in
mechanics problems that David Bowman and I see is to recognize
that the trajectories are unique, and the time evolution of
the motions is just different. Low-g means slower motion,
that's all.
The ironic thing about the problem I quoted is that the person
who made it up felt it necessary to specify the value of g. If
the student is guided by that he will solve the problem somewhat
inelegantly, but I can't make my students (who must already have
done such problems in high school) see that there is merit and
beauty in the elegant solution. It gives insight which the high
school student won't see.
The sliding-off-the-hemisphere problem is very closely related
to the meter stick falling from rest after standing against a
wall problem. I beieve the answers to both problems are
the same, something like arccos 2/3.