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.)
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.
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.)