When did having a pilegesh become unacceptable and why?
A number of reasons are given ranging from practical to hashkafa. One of the main reasons is that the Torah points out that, while allowed, it is not the ideal (as stated in Bereishis 2:24 with Adam). Rav Hirsch goes into detail about this. The ideal is to have the two join together and recreate the unity that existed before the split into Adam and Chava while also fixing the problem of being "alone" which was "lo tov".
A number of reasons are given for Rabbi Gershom’s ban:
It was instituted to prevent people from taking advantage of their wives.2
It was intended to avoid potential infighting between rival wives,3 which may also lead to the transgression of a number of biblical violations.4
Rabbi Gershom was concerned lest the husband be unable to provide properly for all his wives (especially during the difficult times of exile).5
The ban is intended to avoid the inherent rivalry and hatred between rival wives
There is a concern that a man may marry two wives in different locations, which may lead to forbidden relationships between offspring.6
While it has been suggested that it was adopted from Christian practice and laws, to avoid Christian attacks against Jews who act otherwise,7 this argument has been assailed by many other halachic authorities.
As far as Jewish thought is concerned, it would seem that polygamy is not, and never was, an ideal state. The mystical works are replete with references to husband and wife being two halves of one whole. Interestingly, I’ve never encountered an episode in the Talmud or Midrash—which predate Rabbi Gershom’s ban on polygamy—which involves a polygamous family. While it is certainly possible that such stories do exist, it is quite apparent that polygamy was never the norm.
2. Maharik in the name of Rashba, cited in Darchei Moshe, Even ha-Ezer 1:10.
3. Mordechai, Ketubot 291, cited in Darchei Moshe ibid. 1:12.
4. Responsa of Maharam Schick, Even ha-Ezer 4.
5. Responsa of Maharam mi-Padua, 14; Responsa Mishkenot Yaakov, 1.
6. Mishkenot Yaakov ibid.
7. Responsa She’eilat Yaavetz 2:15.