It's very important first to understand that evolutionary processes do not have a causative role in all evolved behaviors or traits. Often species responses to various stimuli are simply that responses to either stimuli in presence or absence and not a behavior that has evolutionary causation.
Grief is a perfect example of such a response. It is a reaction to the end of a continuous stream of signals that otherwise would come from an individuals (or items) presence and represents a momentary set of *shocks* at the re-realization at the loss when need presents.
"Need" can be as simple as the desire to talk to a person who is gone or hear their laughter. Unlike the things we surround ourselves with...which we experience sensorially only in a limited band of ways (and thus impress networks in our brain in an equally limited way)
The influence of individuals we've been close to on our brain networks is extensive...especially when those histories go back to childhood...where dense networks of experience touch upon the active presence of those persons not just in our past lives but in our present minds.
Remember how modulation of our mental state is enabled by the controlled release of various neurotransmitters that maintain the stability of our cognitive landscape through the optimal flow of these brain drugs in order to keep us at balance.
When some one dies we literally have the mental network system thrown out of balance across deep areas of the brain...from autonomics to higher cortical function. Of these people we often have network connections that cross all of our senses from somatosensory to sound to smell to sight to taste...and those often connect to temporally associated events that tie to the personal experience...for example remember the type of perfume some one would wear or remembering the feeling of their hands.
Shakesphere in his genius touched on the core of it when he wrote:
"To die: to sleep;
No more; and by a sleep to say we end
The heart-ache and the thousand natural shocks
That flesh is heir to, 'tis a consummation
Devoutly to be wish'd. "
-- Those thousand natural shocks erupt when we lose some one because they reverberate in a dense multidimensional way in all our mind which our brain must subtract if it is to remain balanced in the face of their absense and it does that by remodulating the brain chemicals...forcing us to prune the mind literally of the old state of expectation that was associated with past interaction with that person to a new state in which we are...less often over time... startled by the sudden realization of their non presence when we think of them. If this subtraction process does not happen the constant shocks can lead to significant build up of stress which could literally kill (and it does quite frequently to those who die from grief complications after the passing of a loved one).
That's the individual side of things but we are now individuals that live in a social structure that also impinges different pressures, for a kin group open mourning (weather the mental grief and detachment process indicated above is real or performed) shows to the individuals in the clan that the individual meant something to the indivuals mourning and to the group...it allows later the group to come together in the common way in which literally they were molded (their mind literally) by the presence of the one departed and allows them to come together to *fill any gaps* that their absense may leave in the survivability of the group. In the past the death of a prominent provider could wreak havoc on an entire groups ability to survive if that person possessed unique skills that the group relied on...mourning enables this rebuilding of a new group dynamic for those roles to be established in the clan so that it can move on.
So here is revealed the true causative drive in my view for grieving...the social import induced as a response to the physiological reaction to loss that individuals must necessarily engage if they are not to die by a thousand continuous shocks.
The hypothesis can be tested by correlation to other species. Contrary to your statements above regarding advance (a term which doesn't make much sense over all evolutionary space because often evolution leads to regression) we should find a correlation between grief and social animals that doesn't exist in non social animals...and indeed a cursory examination of many non social species reveals that they do not mourn. This is not to say that ALL social animals do, many insects indeed live highly social lives but they are cognitively simple beings compared to the complex landscape of sensory experience that weaves together the cognitive states of higher mammals and birds that are social.
So I posit that there is a set of feedback systems at work that feed social evolutionary pressures into expressed behavior depending on the cognitive sensory complexity of the animals in question. The necessary neuro chemical complexity that gives us the greater cognitive dynamism and consciousness also gives us the burden of needing to unbind the deep networks being formed across our sensory landscape with individuals that is continuously Shepparded in the presence of the individuals...once they are gone the energy required to maintain that system in the face of their absence and the shocks that continuously testify to that fact can lead to life ending stress...so to cut the bond we grieve internally (rebalance of neurotransmitters around the kernels of experience that were associated with that departed)...and to acknowledge and strengthen the social web in which we live we grieve externally.
So I assert that Absent social expression in a species one should find that mourning does not present...this is a falsifiable hypothesis that I am interested in testing.
I've not read the literature on theories for the purpose of grief any with information that may illuminate this subject may provide such in the comments below.