This is my take on it.
The purpose of eating is nutrition and health.
And yet I have milkshakes all the time, seemingly without sin (or without inhibiting my union with God)
Milkshakes might not be nutritious, but they provide you with energy your body can use, so a drinking a milkshake isn't sinful, but it can be if you do so for the wrong reasons.
The purpose of my reason is to consider and understand God's presence in my life.
And yet I enjoy sophistic arguments with my friends, seemingly without sin (or without inhibiting my union with God)
Your intellect also helps you to survive. Part of survival is forming social ties with other people. Conversing with friends about day-to-day things is part of learning to love your fellow man, which can bring you closer to god. If the conversation is leading you away from God, like you're talking bad about other people, then it can become sinful
The purpose of my physical health is to serve the less fortunate in the name of God.
And yet I play sports and engage in other physical activity, seemingly without sin (or without inhibiting my union with God)
Serving the less fortunate is part of your duty, but it's not the only purpose of physical health. Sports helps you to maintain your physical health, hence keeping up your ability to use your strength to help the less fortunate. It's not sinful to exercise, even if it's fun or for pleasure, as long as it doesn't interfere with your other duties. Don't play football when you should be at Mass. Don't spend every night at the gym, when you should be spending time with you're family. Do it in moderation, and it's not sinful. Like anything else, do it in excess, and it can interfere with your duty to serve others, subsequently interfering in your relationship with God.
When you have sex and close yourself off from reproduction, it's a different scenario from the examples you've given. When you eat a milkshake, you're not refusing the energy it gives you. When you're talking with your pals, you're not refusing to build bonds with them, or refusing to accept the way it builds intellectual muscles. When you're engaging in sports, you're not refusing to accept the health benefits that go along with it. In fact, you can't refuse any of these things, because it's the natural order of things.
When a couple uses contraception it's interrupting the natural order of things.
It's like drinking the milkshake for the taste then spitting it out because you don't want the calories. Imagine watching someone do that on a regular basis. It's disturbing and disordered--don't you think? You'd tell them, if you don't want the calories, just don't drink it.
It's like playing sports, yet somehow refusing to accept the benefits of the increased strength and endurance it gives you.
It's like hanging out with the guys every weekend, but you close yourself off to their friendship.
It's like inviting your someone to debate you, when you're wearing earplugs.
When a couple has sex they ought to be open to procreation. That's the right reason to have it, even if the likelihood of having a baby is unlikely (due to age or infertility). Having sex is like sending God an invitation to your house. In the case of an infertile couple, there's little chance He's going to knock on the door, but in the case of a couple that uses contraception, they invite God to come, but if He actually shows up at the door, they hold up their hands and say, "Get back, You can't come in!"