Well, that depends on the local climate and the rider's constitution.

But my post was not about the relative merits of cycling versus driving. I simply wanted to rebut clairity's point that making cycling and walking safer necessarily makes driving slower.

Transportation cyclist in Texas here. The need for showering is overstated, I think. Even with the summers here I only know one cyclist who would take a shower after cycling, and they only do so for their training, not their commute as far as I know. If you ride at a comfortable pace and are in good shape, I find that a quick 5 minutes in the bathroom to freshen up is plenty, if it's even necessary. I usually wipe sweat off my head and chest and sometimes will change clothes. Outside of the summer these steps are rarely necessary, by the way.

Also, I think I would frequently (perhaps not every time, but at least 25% of the time) beat a driver to the door of my workplace from my apartment because of a combination of cars not actually being much faster than me due to traffic, stop lights, etc. and parking. My parking is so small that it's a lot closer to my building. This obviously depends on where you work, but here it seems to be a major factor. (On second thought this would also depend on which lot the driver would park in. Most would have a long enough walk to make cycling faster or at least about the same. I also haven't done this race, for what it's worth. Anyway, the point stands: travel times for cyclists are not necessarily worse than drivers.)

By your description, your "5 minute freshen up" isn't included in your travel time...

Generally it's included, especially in travel time. Generally the bicycle wins if you live in a city, even with if you freshen up at home and at work.

I had the time in mind. The ambiguity was a consequence of poor writing. I doubt this takes more than a minute or two in reality, and it's only necessary for perhaps a quarter of the year. Averages to around 30 seconds a day if it takes 2 minutes when necessary.

you really don't need to shower and change your clothes. You can if you want to ride your bike fast and push your physical limits, but generally casual biking won't make you sweat more than if you were walking.

On some days in a dry, warm climate (e.g. California) it's actually a good deal cooler than walking due to the air flow.

Generally, casual biking has little average-speed benefit over walking, in which case why bother?

What do you mean by casual biking? Walking home takes me a half hour or longer. Biking is less than 10 minutes according to my bike computer. Big time difference. Plus if it's hot I spend a lot less time in the sun on the bike, making biking less sweaty than walking. And I can carry a lot more cargo more comfortably.

Biking is still at least three times faster than walking for the same energy input. Even near-zero-effort "casual" biking is as fast as a brisk jog.

You don't need a shower after bike ride on a level surface less then 10 km. Unless maybe you have a heavy headwind.