I agree that sport in schools shouldn't just be a case of shooting the most balls in the net or crossing the finishing line first. That does become very competitive and some pupils don't stand a chance. There is usually something that an unsporty child can do well. Schools need to broaden the sports curriculum to include all disciplines, but money and resources is a problem.
At the schools I teach in, sport extends to cheer leading and jazz dance, swimming, canoeing, cycling proficiency etc., which as mentioned above by Aveline, become more like hobbies.
My reservation with streaming pupils in Sports' lessons would be that you would then have to stream them for Music or Art as well to make it fair.