It is a fair bet to say that few military members looking for an easy way out of their service obligations following enlistment choose to apply for discharge as a conscientious objector.
Here's the reason why: It is flatly hard to be awarded that status. One online overview of discharge requirements for conscientious objectors calls the application process "challenging," noting the "many pitfalls to be avoided" at various phases of the process.
In some instances, it might take an objector a longer time to come to a firm realization regarding his or her beliefs, act on them and exhaust all relevant military processes than it would take to simply serve out a military term.
One central reason for that is because military authorities must be satisfied that an objector claim is truly based on the "fixed and firm" beliefs of a servicemember, rather than on a whim, sudden state of mind or strategic ploy to beat the system.
In other words, a military member must demonstrate a singularly strong opposition and aversion to participation in warfare and in working to promote military aims. That opposition must ring loud and clear in detailed application materials and additional evidence that might conceivably be considered at both a hearing and appellate level. Additionally, input from both a psychiatrist and a chaplain is required in the process.
Understandably, there is some subjectivity in the phrase "fixed and firm," with room for creative arguments on all sides of the question. As noted in the above-cited overview, "a member considering making an application [based on objector status] is well advised to obtain qualified legal advice."