[Note: I shifted the question from "children shooting in schools" to the broader category of "mass murder". I think this is a reasonable jump, but it may be a weakness of this answer.]
Back in 1998, the question of what motivated mass murderers wasn't well resolved, but one model of motivation included at least five elements.
Multiple Homicide: Patterns of Serial and Mass Murder, James Alan Fox and Jack Levin, Crime and Justice 1998 23:, 407-455
Importantly, the difference of timing that distinguishes serial from mass murder may also obscure strong similarities in their motivation. Both can be understood within the same motivational typology-power, revenge, loyalty, profit, and terror. The research literature, still in its infancy, is more speculative than definitive, based primarily on anecdotal evidence rather than hard data.
By 2016, there had been progress, but it still was an open problem.
Much speculation has been made in the media as to the causes of mass murder in the United States, yet little empirical research exists to verify factors leading to violence.
This paper attempts to address that by looking at media reports from 152 mass murders from 2007 - 2011.
The author considered a number of typologies they found in the literature, including Fox and Levine's:
power (e.g., mission to change the world), revenge, loyalty (e.g., husband kills family during financial hardship), financial profit, and terror.
family annihilators where the male head of household kills because he is “depressed, paranoid, intoxicated,
or a combination of these” [..]; pseudocommandos who enjoy guns; and
set-and-run killers who commit their offense in a manner that allows for them to
escape [...]; disciples (e.g., killing for a leader) and disgruntled employees.
anger/revenge, domestic/romantic situations, interpersonal conflict, and crime-related, gang-related, political, and a nonspecific motive.
For this particular study, the author used:
emotional triggers (i.e., the
loss of a relationship, the loss of a job, a fight, or “other”), general relationship/domestic issues, financial issues, mental health issues, criminal gain, and political motivations (e.g., terrorism). Although anger and revenge have been used in prior typologies (Fox & Levin, 1998; Petee et al., 1997), these motivations were not included in the current study. The reason for this decision was because all mass murderers, identified for the study, who were motivated by anger or revenge had some precipitating event (e.g., divorce, affair, loss of job) that resulted in the anger and desire for revenge. It
was believed that the catalyst for the event was more important for determining motivation than merely identifying anger
Note that these are not mutual exclusive.
In conclusion, there are several different ways of identifying the motives for mass murder. None of them is right but some have been useful. Of the examples I found, the three-element model in the question is the most simplistic.