It seems that there are some contradictory assumptions in the Dalai Lama's presentation of his logic on the ethic of compassion, granting that he may be speaking in simpler terms than those in which he actually thinks. The underlying assumption of "universal responsibility," which is later equated by definition with "universal compassion" and by requirement with "spiritual democracy," is that even one's enemies are motivated by thier "quest for happines."
Later, it is established that "universal responsibility" requires personal sacrifice, one assumes a sacrifice of one's pursuit of happiness, and the "neglect of egotistical desires," again, one assumes desires for happiness. Thus the proposition that all are motivated by the pursuit of happiness is undercut and invalidated since at least those who agree with the presupossition of personal sacrifice and neglect of egotistical desires are not motivated by the pursuit of happiness.
Futher, it is asserted that "all beings want the same thing we want." If not all want personal sacrifice and the neglect of egotistical desire; conversely, if not all want personal happiness that comes regardless of the cost to the "common good" or another individual's good, then, once again, the presupposition that all are motivated by the pursuit of happiness is undercut.
This leads to the realization that the term "happiness," as acted upon in the contemporary milieu, is inadequately defined in the context of "universal responsibility/compassion/spiritual democracy" and doesn't incorporate the realities that make distinctions of personal sacrifice and neglect of egotistical desires a requisite choice for demonstrating compassion, peace and love.
Therefore, it seems necessary to conclude that, no, not all people want the kind of compassion, peace and love advocated by the Dalai Lama in his statements on the ethic of comapssion. It must be concluded that some individuals incorporate selfishness (the opposite of personal sacrifice) and egotisical desire that does not weigh the cost of its fulfillment into their individual definitions of "happiness."
[You don't identify the source you read from, but the source I read from in aswering this question is "The Dalai Lama On Compassion" at His Holiness the Dalai Lama of Tibet.]