The explanation you heard about why specifically the man is obligated is one of many possible explanations. Regardless, your question is an excellent one. Once the couple has a boy and a girl, they have fulfilled the Biblical command. At that point, it is very common, though not universal, for Rabbis to give a heter (permission) for the couple to practice birth control for up to two years. Since the mitzvah is not just to bear children, but to raise them, emotional and economic factors must be taken into account. Many rabbis will extend a heter beyond two years in light of these factors.
A rabbi should still be consulted, even after there is a boy and girl, because of the rabbinic admonishment to continue to procreate even after fulfilling the biblical quota. This admonishment is expounded from the verse in Ecclesiastes 11:6 "In the morning sow your seed, and in the evening do not desist." In the Talmud (Yevamot 62b), R. Yehoshua explains that a man should have children both in his youth and in his old age, and the Shulchan Aruch rules according to his opinion (Even HaEzer 1:8). Although the exact definition and parameters of the rabbinic obligation are the subject of much discussion and debate among the prominent halachic authorities, most agree that one may not decide to cease and desist from procreating forever. Our site's rabbinic supervisor, Rabbi Yehuda Henkin, has argued that a couple has fulfilled even the rabbinic command once they have two boys and two girls.
The question remains: what happens when a couple seeking a heter to use birth control after having both a boy and a girl are turned down? The classic rabbinic approach was only to permit contraceptive use when a woman's life was at stake. Since the classic rabbinic literature on this topic precedes the availability of the contraceptive pill, we are left with a quandary: how much of the rabbinic reluctance was based on the methods historically available and how much was based on timeless Torah principles? A similar question can be asked of our own attitudes: how much of our desire to take birth control is a function of the prevailing culture and how much of it is truly reflective of our attitude towards mitzvot in general (embracing them and not desisting from them)? The wide range of answers to these questions among rabbis and couples is connected with the range of halachic answers on this issue.
Ultimately, what is most important is that a couple be familiar with a rabbi's approach to these issues before seeking his counsel. A couple should only seek out the counsel of a rabbi whom they respect enough to be prepared to honor his evaluation of the issue. The rabbi, in turn, must learn enough about the couple's situation to apply his approach appropriately and to explain the factors involved in a way the couple can understand and accept. In this way, it is less likely that a couple would receive a psak (ruling) that they could not deal with. If they could not deal with it, communication would be strong enough that they could contact the rabbi again with their concerns. A rabbi takes the urgency of a questioner's concerns very much into account.