If a priest who has had five years to discern his vocation can be laicized, why does a couple, who may court only six months to a year need a Church annulment? Doesn’t a person who wishes to remarry deserve happiness?

Your question seems to imply that laicization is a simple process. It is not, and often takes investigation, the preparing of a petition, and sometimes the gathering of testimony. This may take years to complete. Annulments, while not easy, can often be accomplished in six months to a year, depending on the diocese and complexity of the case. But the fact is, neither are easy.
 
That said, there is an important difference. Laicization does not generally seek to prove an ordination never took place or was invalid. It presumes the man was validly ordained and only releases him of his ecclesial obligations to live all the disciplines of priestly life such as perpetual celibacy, and the duties of the saying the Liturgy of the Hours, and celebrating Mass, etc.
 
Annulment on the other hand, is the recognition by the Church, based on evidence given, that a valid Catholic Marriage never occurred, since something essential was lacking. This of course requires proof that must be presented and then considered. And that, like the process of laicization, takes some time.
 
Both processes ultimately involve matters of great sadness and have significant pastoral implications. For while recognizing human struggles, the Church must also seek to uphold the gravity of vows that are made. Showing compassion to the individuals who seek annulments or laicization must be balanced with the common good, the reality of sacraments, and what Scripture teaches. The happiness of certain individuals cannot be the Church’s only concern. Hence, the pastoral process involved must necessarily be thorough and careful.