The English phoneme typically represented by the letter ⟨r⟩ represents a confusing and complicated mess of allophonic realizations, some of which are highly disparate and some of which vary only slightly. One feature which is common, especially in American English, is some degree of lip rounding on the sound, which is common indicated by a labialization diacritic: ⟨ʷ⟩. I think that my own pronunciation (Midwest) is molar and moderately rounded: [ɹ̈ʷ].
This lip rounding does not seem to have any significant auditory effect on the sound unless the rounding is unnaturally severe. Thus, I am wondering if there is an explanation for it, and I have a personal theory.
Babies and young children often have considerable difficulty producing the approximant /r/ sound, as it is infamously one of the hardest sounds in the English language. This leads many young American children to pronounce an entirely different phone which is very easy for them, often some sort of glide like [w]. Eventually, most of them learn to place their tongue in the correct position to make the standard liquid sound.
I think you might be able to see where I'm going with this. If children are realizing /r/ as [w], even when they learn to place their tongue in the correct [ɹ̈], [ɹ̠], or [ɻ] position, they may keep the labialization of the [w] as a leftover from their early years. From generation to generation, this leftover rounding could have become more and more common to the point that it is now standard in American English.
I have never seen this explanation discussed anywhere, so I have no idea if it is true or not. Is there any merit to my theory, or is there some other reason for r-rounding? Have similar phenomena been observed for other sounds? I should note that this could actually go entirely the other way - maybe babies make a [w] sound because they already see their parents using a rounded realization, and not the other way around.
*Note - What I'm talking about is not rhotacism, which involves pronouncing /r/ as a labiodental approximant, although the existence of that phenomenon is probably due to a similar reason.