I wear a scarf or pashmina pretty much every day, in hot and in cold climates and absolutely always when flying or taking the train. I find it's mainly US airlines which don't have pillows or blankets, so I can use my scarf/pashmina instead. Or when flying LH, which is always too warm, I use my scarf/pashmina instead of the LH blanket.

I have no concerns about germs, however, on airline pillows/blankets/seats.