I can cycle hard and sweat, or I can cycle gently and sweat very little. We don't have a shower in the office, but I don't cycle too hard to work, and change shirt when I get in. It's not a problem. On the way home, I cycle hard and have a shower.
I can stay completely dry when cycling by simply carrying waterproofs. I just make it routine to always have these. It's true that these make it a little sweatier, but a change of clothes solves this problem.
Cycling isn't going to be suitable for every journey. However, 15% is quite a small target when you think about all the little trips people need. Things like cheap electric cars can be useful for the remaining 85%.