What is the origin of the phrase "hurt(s) like a bitch/motherf***er/bastard"?

The first two cursewords are specifically and characteristically common to modern black American vernacular English, as is the dropping of the "s" in the word "hurts," so that is strongly suggestive of some relationship to this phrase (note also that your Charles Davis citation is actually from a book called On Being Black).

I'd hazard a guess that the increased prominence of the phrases' general usage starting in the 60's might be related to the increased racial integration of American society, and the resulting more powerful and less filtered cultural influence of black America (compare the dates of the introduction of "blaxploitation" films). Anecdotally speaking, these words are more common in mainstream depictions of black speech than in actual black speech, which suggests a particular fondness for them as signifiers of blackness in mainstream culture.

In terms of usage, I read them as first anthropomorphizing the pain, and then cursing it. Anthropomorphic language is also characteristic of black American vernacular. It's worth noting that "bastard" is an outlier here, while anthropomorphic, it is not specifically characteristic of black vernacular.

Two possibilities I can see:

1) It is actually referring to the pain caused by a "bitch", which one could take to mean "A woman who has hurt me, eg broken my heart, not done what I wanted her to do, etc". In other words, hurts like that special way that can only be done by someone with whom we are in a relationship. Note that I am in no way endorsing the use of the word "bitch" here.

2) It's just a general expletive, without much real meaning, like saying "Hurts like a motherf*cker", "Hurts like a bastard", etc.

If someone is in pain, and trying to describe it, they might say "Hurts like a .... (I have no frame of reference as this is the worst pain I ever experienced, in fact it's so bad I'm just going to swear now)... bitch"

I'm not sure how one would go about working out which one was in the mind of the originators of this phrase.

I only quote from memory here: But curse words in all languages have a certain similar visceral effect.

They usually are an affront that you in your right mind being polite would not do.

fuck --> low level bitch --> see fuck bastard --> see bitch

They are all related to sexuality which is what everything revolves around anyways.

Why we curse

Like what DisplayName said, cursing relieves stress and has similar effects. Pretty much anything forbidden by society (verbally) creates appeal in using them.