Not really. In the first place, stating a prime purpose does not preclude other purposes, unless they are positively excluded or would frustrate that purpose.
That depends on the basis upon which the preclusion is claimed. Specifics matter. A citizen's right to life may be lawfully infringed under specific circumstances, but
all other purposes government may assume or assert, relative to that right, are precluded, no matter how needful they may seem at a given time. Likewise with all other natural rights. Else our claim to rights, and to a government that secures them, is wholly academic—and meaningless.
Secondly I should think right to life and pursuit of happiness would cover a great number of things that people need to access in the modern world,
We may think that, but
do they cover things simply because they are claimed to be needful? For example, you may claim to need housing, but nature affords you no right to housing, only the right to pursue it. Government secures from infringement your right to pursue housing, but it cannot secure for you the right that doesn't exist—the right
to housing. You may claim to need education, but nature affords you no right to education, only the right to pursue it. Government secures from infringement your right to pursue education, but it cannot secure for you the right that doesn't exist—the right
to education. And so on. Government secures one's
real natural rights; everything else is up to the individual to obtain using his natural rights. Every imposed deviation from this results in injury to someone in some way.
and perhaps even libraries, not to mention social security, health services and so on.
No one has a natural right to social security, health services, libraries, etc. How can government secure a natural right that doesn't exist?
Don't get me wrong here; I'm just looking at the law. I
personally want everyone to have every service and amenity he needs in life. And where he can't secure a needful thing for himself through the deployment of his rights and abilities, I believe he should be helped by his fellows, which includes me. But government has no lawful authority to compel my neighbor, who doesn't agree with me on that point, to surrender any portion of himself to support my belief. If it does, government is a tyrant, claiming unto itself all powers of justice (state) and mercy (religion). It becomes the fox guarding the hen-house, so to speak.
To take you literally, one would think the government has no business offering schooling.
That is correct; government has no lawful authority to impose education on its citizenry. The people yield to this affront, but government has no lawful authority to impose it. Education is wholly a religious question, which places it entirely outside the purview of government. Additionally, rights and obligations associated with a thing belong to the creator of that thing. Parents create children, not government, so the right to educate belongs to the parents,
as do the obligations of defining, funding, and deploying that education.
What does government create? Over
those things it is sovereign—and over nothing else. When government starts producing children, government may justly define education for those children, and fund their education through the fruits of its own productivity (not mine; the fruits of my productivity are mine to deploy to educate
my children).
But, to get back to my original question, in what sense are these sports leagues imposed?
In the sense that government unlawfully appropriates a portion of our natural rights to provide the service.