Get just a little smaller, and dedicate a (grade-separated) lane to them, and you have bicycles in a bike lane.
E-bikes mostly solve the effort and sweat aspect. But that introduces a larger entry cost.
I'd say distance is the biggest factor preventing it taking off in the US (Outside of Cities). A mixture of cycling + transit could solve the longer distances.
Even in "rainy" London, it's only a small fraction of days that are like that. But you need a solution for those days, too.
Cycling is an excellent mode of transport in cities, but it's not perfect. Some sort of covered bike or micro-car would solve some of those problems.
And the solution is easy: a bike with full length fenders, a poncho, some gloves, possibly some change in an hermetic bag.
Also rain in summer is not an issue, sometimes it is even pleasant and refreshing.
And you probably also forget that in most motorists case, they don't all have the change to have an indoor parking both at home and at the office / place they do errands. Most need to walk from home to parking space then from parking space to office/shop...and get soaked in the process.
You're not made of sugar. You won't melt in the rain.
Majority of Americans will disagree is what I'm saying. We're used to climate controlled boxes, this allows people do get dressed up for things and wear a bunch of make-up/hair product, etc. Not everyone, but a majority. Culturally, when I travel it's a thing that I notice a lot. Other places that use bikes and have that cultural norm, they also tend to normalize the other dress code parts that go along with it. In US, it would be a tough sell because we have nearly no basis for this concept. Even people that go to a gym tend to shower and change clothes (including grooming) prior to doing anything else social/work related.
Now, seriously, we don’t get much snow (to the point 10cm halts a city) but we do more than a fair share of cold rain and gusty wind.