Celibacy can be traced back to the very earliest Church. It has become a discipline in the Roman rite, that a married man may not be ordained; but it is not an absolute discipline.
There are a number of issues at play in any conversation about celibacy, and they need to be sorted out and kept separate.
There seems to be a perception among some (at least, some of the comments and pronouncements out of Rome seem to indicate it) that there is some move to “do away with” celibacy. I do not know of any such proposal; but when the discussion comes up officially, that seems to be lingering about the response.
It has been suggested that the discipline be relaxed as to priests (the Roman rite allows for the ordination of one to permanent deacon who is already married), and Rome’s most recent response is that it is not going to happen. That does not mean that it could not happen in the future.
The comment often comes that if we allowed married men to be ordained, that the shortage of priests would be lessened. Actually, it appears that the shortage of priests is not necessarily a world-wide phenomenon, but rather seems to be more the issue in Europe and North and South America. And the cause, or causes of the shortage in those countries seems to run to issues more of secualrism, loss of faith (in the US about 1 out of every three Catholics attend Mass on a regular basis, and 50 years ago it was about 7 to 8 out of 10; Europe in many areas is down to 1 out of 10 or fewer), lack of catechesis, and lack of willingness to promote the vocation. That is starting to change in some areas but it is slow.
It is also noted that Protestant ministers numbers are down, and they allow married men to be ministers; so marriage in and of itself may not be the issue.
The Roman rite does allow married men to be ordained, but they are converts; mostly from the Episcopal/Anglican group, then Methodists and Lutherans, and there is one former Presbyterian (who is in my Archdiocese).
Much is made about priests not being able to serve their parish; that is a slap in the face to all Eastern rite priests who are married. It is purely a straw man arguement.
Celibacy is an identification with Christ, but is not intrinsic to the priesthood; the Eastern rites have ordained married men since the founding of the Church and its divisions into various rites.
It is not, however, a panacea to perceived ills - shortages or other ills. It is a separate vocation from the priesthood. but at this time in the Western rite, anyone who has not come from another Church (e.g. Anglican) will not be ordained if he is married.