Shouldn’t that make you sick for longer because your body can’t fight as well?

Short answer: It's not necessarily true for most over the counter medication, with some exceptions.

Your body should have no trouble creating antibodies to fight off the infection because those drugs don't act on white blood cells. And at around the three-to-four-day mark, the infection is almost guaranteed to go away on its own when the adaptive immune response kicks in.

Interestingly enough, most of the cold symptoms (except for fever) are part of the viral strategy to help it spread. In a weird way, the infectious agent kicks up your primary immune response to do its bidding.

What's not recommended are fever-reducing medications. This is because elevating body temperature is your body's way of reducing growth rates of most microorganisms and also speed up its own attack and alerts its own defenses. Reducing that temperature has a chance of increasing viral shedding. Ironically enough, this list of antipyrietic drugs includes Aspirin and paracetamol which are also used for pain relief. lol

My personal take is that it doesn't matter much unless you're overly concerned about being sick for--I'm guessing--half a day to a day longer with subsiding symptoms, and are overly worried about increasing the average rates of infections by a marginal amount.

(Full disclosure: I'm not a medical doctor but a former pharmacologist.)

Your body's responses are overkill/unnecessary in a lot of cases. Like inflammation, a lot of that is so your skin will swell up near a cut so not as much dirt gets in it, but if you're keeping it clean/bandaged that's probably counterproductive. A runny nose, isn't that just the virus tricking you into becoming more contagious? A fever, could be useful if it's a virus that can do serious damage before your immune system can ramp up enough to handle it, but if it's a less dangerous one that's probably just self damage.

ibuprofen is just actually magic