IMHO, a windshirt is a shirt, but it can certainly have the nuances of a jacket– full zip, hood, etc. Think "high performance long sleeve shirt" rather than "jacket" and you'll see the value.
It should prevent convection of your body heat to the colder air around you. In addition, it can keep wet brush off and provide some protection from insects (get a light color).
It should be comfortably breathable. A windshirt that is not breathable is useless: you might as well have your rain jacket on.
It may give some short-term protection from light rain.
A windshirt is not rain gear and should not be depended on to stay dry in heavy rain.
Carrying both a wind shirt and a button-down shirt would be redundant. Most button-down shirts are heavier than a high performance hiking windshirt anyway.
A light rain shell can certainly be worn like a windshirt and will protect from wind, but it won't be as breathable. And a minimalist rain shell is typically missing the venting options that will make it closer to the comfort of a windshirt.
Using a windshirt and insulation layers with wind-resistant properties is problematic: there is some redundancy. Wearing your windshirt over your down insulation will provide another air space and help protect the down from light rain, but that might also trap moisture in the down, so it depends on the conditions and your level of activity. If it is so cold that you need another layer with the down, you might want to wear the windshirt under, precip aside. What works for me is a windshirt over an insulating *vest* and long sleeve base layer. That gives maximum bang for the gram.
There's no free lunch. What is needed is a good breathable windshirt that has minimal weight to offset any redundancy with the insulation and rain shells. If you can hike comfortably in colder air and wind, then a windshirt may be extra gear and you don't need it. If you find yourself cold while underway with just your base layer shirt on, then the windshirt is valuable, quite literally "worth the weight." In camp you can wear your insulation layer or your rain shell and get the same protection that the windshirt would provide. I do often putter around camp with base layer and windshirt, just like wearing a tee shirt and long sleeve shirt at home or work.