Why do Muslims stick their feet in the sink in public restrooms?

It's called 'wudhu'/ablution and it's about cleaning up/purifying one's body before standing for prayer, not the other way around as bystanders might think!
I thought of putting up this post when I recently heard people denouncing the ritual (especially when performed in a public restroom) as ''gross'' and ''unhygienic''.

The method of performing ablution comes from directly from the Qur'an:

''O you who believe! When you prepare for prayer, wash your faces, and your hands (and arms) to the elbows; Rub your heads (with water); and (wash) your feet to the ankles. If you are in a state of ceremonial impurity, bathe your whole body.

But if you are ill, or on a journey, or one of you comes from offices of nature, or you have been in contact with women, and you find no water, then take for yourselves clean sand or earth, and rub therewith your faces and hands, Allah does not wish to place you in a difficulty, but to make you clean, and to complete his favor to you, that ye may be grateful.” (Chapter 5, Verse 6)

Before a Muslim performs his prayers, he carries out the ablution movements mentioned in the above verse as well as others, which the Prophet (peace and blessings of Allah be upon him) has added.
Thus, the ablution comprises of washing the hands, arms right up to the elbow, face, mouth, nostrils, and feet up the ankle, all three times each. The inside and behind the ears, as well as the part of the head above the forehead is wiped once.

Done five times a day, it not only cleanses these vital parts of the body from dust and dirt but also "softens" and refreshes them.1
Interestingly enough, the Prophet (peace and blessings of Allah be upon him) also encouraged doing ablution before going to bed. This same ritual is also encouraged by Yoga experts who say that washing important motor and sensory organs such as the hands, arms, eyes, legs, mouth and genitals before sleep using cool water relaxes the body preparing it for a deep sleep.2

Read the complete article here

So what happens when it's time for prayer and a Muslim's in a public place -- should they simply stick their feet in the sink and ignore all the horrified/disgusted looks they get?

This is being touted as an option:

I'm not sure how feasible / acceptable this would be, but speaking for myself, I'd rather perform my ablution in peace, somewhere private (since it is an act of worship as well) rather than show complete unconcern for other restroom users and perform it in full public view, knowing fully well that it might seem off-putting (to put it mildly) to some people.