We do not live in a mile climate. We are in Kansas City. It can be 60 and raining one day, and the next day is 30 degrees with snow and ice. The black paper (whatever it is) was put on and it was torn all off by the wind in a matter of a week. We are also supposed to get another foot of snow in five days. Should I just take lots of pictures and demand that they replace the entire sheathing when it finally melts in a month? Obviously they neglected our roof. I do not see any other houses building built around us with just exposed sheathing.

First, find out what the roof sheathing is. AdvanTech is made by Huber and they also make the green Zip wall sheathing so its highly likely it is AdvanTech and designed for weather exposure.

OSB swells when wet and doesn't completely shrink when dry but AdvanTech OSB is designed to not do that.

For AdvanTech there is a 500 day warranty for weather resistance during construction.

Second, the builder intends to replace the black asphalt "building paper" underlayment when the roof is dry but you should insist that the entire roof be covered Grace Products "Ice & Water Shield". That is what will be used at the hips, valleys and eaves anyway so why not over the rest of the roof? I design for MA weather and its pretty standard practice here.

I wouldn't worry too much about it (no, you're not getting new sheathing). As RES stated simply find out what product was used on the roof and go from there.

I wouldn't call roofing felt "archaic". It is specified by many different manufacturers....

"Minimize exposure to weather by applying shingle underlayment or #15 roofing felt (use #30 for extended exposures) that meets ASTM D 226 or ASTM D 4869 standards. Minimize buckling by allowing the sheathing to reach ambient moisture conditions before installing roof coverings. See page 7 for additional information on buckling. Use heavyweight or laminated and textured shingles to help hide any imperfections in the roof and to give the best appearance."

"https://www.weyerhaeuser.com/woodproducts/document-library/document_library_detail/osb-4004/

Asphalt felt is no longer made from cotton rags and #15 no longer weighs 15 lbs/100 s.f. Here is what is available today:

So, today this product is nothing like what it was when it became the standard for roofing underlayment and there is no way to even know the weight and many manufacturers don't even meet the ASTM standards. Therefore, no one should be surprised that asphalt felt underlayment was blown off of a roof in Kansas City in the winter. This stuff provides little improvement in waterproofing and wrinkles when wet so in a high wind, cold climate its simply inadequate.

Asphalt roofing shingles became lighter when they switched from organic to fiberglass mats. The same can be true for asphalt building felt but the above ASTM standards are for organic felts which are what is used for steep roof underlayment. Asphalt glass felt is rated by ASTM D2178 / D2178M-15a in lbs. of breaking strengths and are used in built-up roofing.

There are many synthetic roofing underlayments and almost any of them would be tougher than asphalt saturated "building paper".

"Tougher" may not mean better when it comes to weather barriers. We don't know if OP's roof is vented or unvented nor what type of insulation. Obviously, they did inadequate job on fastening roofing paper. Tell your builder/roofer to invest in a button cap nailer for goodness sake.

Fully adhered impermeable roofing membranes make sense in the right applications. We are doing it on our current renovation with board sheathing in a climate with little ice damming concerns. Grace Ice and Water along with most other "peel and stick" options available in US are vapor closed, which allows practically no drying through membrane. Regular building paper could be the better option because it is vapor permeable.

If you look through the various roof stackups in this BSC document, you will find most include a layer of vapor permeable roofing paper. Projects using vapor impermeable membranes over unvented roofs with air permeable insulation should consider vapor diffusion ports.

Wondering how all this turned out for you, as we are in Michigan and are in the same situation. Roof decking went up on Christmas Eve. On the 26th, it poured nonstop for a full day. Rained on and off for a few days. Sun came out and began drying everything, and now it is covered in snow. So stressful!