The consensus among most psychologists who have studied absolute pitch is that it usually requires some sort of musical training before the age of 5-6. There are rare instances of children acquiring it a bit later. I also remember reading of a case of an adult who had absolute pitch but did not have any musical training beforehand. (He had to be tested using a different protocol since he didn't know about note letter names, but otherwise showed pitch identification abilities similar to musicians with absolute pitch.)
Basically, the research consensus is that usually children acquire the standard kind of absolute pitch during a so-called "critical period" where musical training or at least note-naming exercises are usually started around ages 2-4. (This idea of a "critical period" is common in other tasks, like the ability to learn a language without an accent is much harder once a child gets past a certain age, as young children have a better capacity to learn new phonemes, but older children are more "stuck" with the set they've learned.)
What you're describing about being able to replicate the correct pitch of a well-known piece of music consistently is not usually what researchers call "absolute pitch," though it may be somewhat related. Many if not most adults actually have the ability to recall familiar songs very close to the original pitch. This is a standard pitch memory effect, but it generally does not imply the ability to do more advanced tasks that those with normal "absolute pitch" can do, like instantly associate a given stimulus with a specific note name. While there is anecdotal evidence of adults gradually acquiring absolute pitch abilities with a lot of effort, there is some controversy among researchers about whether it can compare with the ability when learned in the critical period of childhood around 2-5 years of age.
However, when you begin to discuss whether absolute pitch is actually a more "evolved" skill or not, I think you may be hinting at the so-called "unlearning theory" of absolute pitch. Basically, it's the idea that all humans are born with some sort of "absolute pitch" ability on a basic level, but we actually "unlearn" it during childhood in favor of the more useful skill of relative pitch. The "unlearning theory" isn't a common view among absolute pitch researchers, but there are a minority of psychologists who hold to it.
Personally, I think the unlearning theory is pretty plausible, as there are various studies on animals that show they have some version of absolute pitch, at least in the sense that they can be trained to respond to stimuli within a narrow frequency range. And the basic recognition of a frequency/pitch is a more rudimentary cognitive process in general compared to a skill like relative pitch.
One might think of an analogy with color recognition. Humans all seem to be born with some ability to recognize color (excepting rare forms of total color blindness), which is effectively a visual frequency distinction. But small children need to be taught color categorization in order to understand which frequency bands correspond to which color names. Different languages may subdivide the spectrum differently, causing children raised with different color names to vary in their speed and ability to differentiate color.
In any case, even without training as a small child, there are strong cognitive and even survival advantages to color recognition. Blood is red, for example. Noticing these sort of consistencies in the environment will happen whether or not language is introduced for color differentiation. However, the ability to understand how colors may be related, say, artistically -- that's a much more higher-order skill that few people achieve.
Turning back to pitch recognition, there's not really a similar survival or environmental need for absolute pitch recognition. It's not like many naturally occurring things always make the sound of a B-flat or whatever. Of course absolute pitch on a wider scale is important -- to be able to tell whether a sound is likely coming from a man or woman or child, for example. But to be able to discriminate pitch identification on a semitone level occurs in fewer contexts. So perhaps small children exposed to music and asked to do pitch-naming tasks are usually the only ones who develop that skill further. There's further circumstantial evidence of this in that absolute pitch incidence is higher among groups of people who speak "tonal" languages where the rough absolute pitch of a word can be important to its meaning. That may indicate that there may be a much wider group of children capable of acquiring absolute pitch, but only some actually develop it.
Meanwhile, relative pitch relationships tend to be a strong part of our culture otherwise. Even in most spoken language, a rise or fall in pitch carries semantic meaning relative to the previous pitch, not to a specific absolute pitch. Musical training in most cultures also tends to focus on the ability to relate pitches relatively, where a given song retains its identity regardless of key/pitch: only the relative pitch matters. Children who are exposed to such stimuli and taught song X in the key of C is "the same" as song X in the key of F will thus quickly learn that absolute pitch isn't a very useful skill in many contexts.
To come back to your question and summarize (TL;DR): there are some absolute pitch researchers who think all people (or at least most) have the ability to acquire absolute pitch, but they lose the ability to do so if they don't have training before around age 5. Other researchers estimate the capacity to acquire AP is less common but still quite possible for many people if they learn it during this critical period of early childhood. Your ability to remember the pitch of one specific song or musical piece is actually a distinct (and very common) ability that most researchers wouldn't say qualifies as "absolute pitch" as commonly understood, nor does it seem to help much with acquiring AP as an adult.
As I said, it sounds like you're interested in the so-called "unlearning theory" for AP, so you may find more information on your hypothesis by searching for that topic. It was originally proposed by Abraham (1901) but promoted more recently by Dixon Ward. Here's a good intro to some of your questions about AP.