In general all these cold and flu medications do is suppress your immune response
That's not true at all. Over the counter cold formulations contain one of four products: decongestant, cough suppressant, pain reliever and antihistamine.
None of those suppress your immune system.
EDIT: I think know what you meant now. You mean your immune system reacts to a cold by making you cough and the drugs suppress your cough (thus your immune response)? In that case, I see your point.
Yes, they are. Also, I believe most pain medications (apart from opioids) work by knocking out some part of the immune response.
Now, even assuming that the medication doesn't actually interfere with your immune system and just gets rid of the symptoms and makes you feel great, you should be aware that your immune system deliberately makes you feel crappy ("sickness behaviour") so that you'll rest and give your immune system a chance to kill the infection. This is more an issue with the flu rather than the cold.
I think what threw me off is the use of the word "immune suppression". That word has a specific meaning when referring to drugs and tends to refer to drugs that to interrupt immune response far upstream. These drugs have severe side effects as your body can't fight invading viruses and bacteria.
I think the difference with cold remedies is that they tend to interfere with immune responses that are far downstream that tend to be more symptomatic, rather than major mechanisms by which the body's immune system works.
That's not true at all. Over the counter cold formulations contain one of four products: decongestant, cough suppressant, pain reliever and antihistamine.
None of those suppress your immune system.
EDIT: I think know what you meant now. You mean your immune system reacts to a cold by making you cough and the drugs suppress your cough (thus your immune response)? In that case, I see your point.