Is your commute that long that it's impractical to just cycle at a comfortable, non-sweaty pace?
When it is not hot out (like now), I'll just wear appropriate pants and skip the shower. I find my dress pants rub too much in certain areas when on a bike so I still have to change, but it is quick and I shower at home.
I have to say that I really love biking to work. It's way less stressful than sitting in traffic and there are some health benefits too. Plus it has a way lower carbon footprint.
Also, I think your advice to ride at a moderate speed is a good one. When I first got into bicycle commuting, I was reading the bicycle subreddit and I was told more than once that it never gets easier, you just get faster. Well, they were wrong. It gets way easier. I understand what they mean by "you just get faster", but I think it's potentially discouraging to new out-of-shape riders like myself.
It gets way easier and for me is almost meditative. My mind wanders like crazy.
Or go the Chinese route and get an electric assist bike.
The bigger obstacle for where I live would be 90F temperature and 80% humidity. You get sweaty regardless of hills (or any physical exertion for that matter) - just by being outside an air conditioned room for more than 10 minutes.
Biking in a hilly place like San Francisco however, there is no escape from being sweaty.
Obviously some people live and work in areas where a sweaty climb is unavoidable, but far fewer than you seem to imply. And it helps a lot that it's virtually never hot here.
Having moved from Georgia, it doesn't matter if the route is flat when it's 95 and nearly 100% humidity. That's way more relevant to sweat than the SF hills are.
For a grade over a certain percentage and longer than some duration, it is probably indeed the case that sweating is unavoidable. But you can usually trade a short steep grade for a longer shallower grade, even in San Francisco. The absolute worst portions of SF hills are usually only a block or two long and are much more likely to be found where you live, not where you work.
You can get just as hard of a workout as with a manual bicycle — it's just that you go faster and cover more distance.
However, it also gives you the option to pedal lightly and rely on the motor more, when getting sweaty isn't convenient.
In my experience, it's an extra 5 minutes of work. Do you consider that to be "a lot"?
Does it really have to? I'm sure you can avoid that by not using hot water (or even going with cold-ish) and not spending more than 3-5 minutes in the shower. Also you can turn off the water if washing your hear takes time. Seems simple to address.