I think you have it backwards. Complementarianism usually allows women to serve, but not lead. This is the position that Paul writes from in 1 Tim 2:12. However, he also outlines his reasoning in chapters 3 and 4 when he says that leaders, overseers, and deacons should not be recent converts to Christianity. They should have extensive scriptural knowledge...
The problem is that the Ephesian women typically weren't very well educated because in 1st century Jewish culture, they weren't allowed to study. Therefore, they didn't have the solid scriptural knowledge to teach. Paul knew there were a many problems with the Ephesians being lured into false teachings, and didn't want it to hurt the fledgling church at Ephesus. So he said the women that didn't yet have the deep knowledge shouldn't teach.
Yet being the operative word in that last sentence.
Paul went to Ephesus at least two other times, and taught there for three years before he wrote his letter to the Ephesians. In Ephesians, he wrote about submission to one another (chapter 5) and unity in the body of Christ, regardless of gender, social rank, or nationality (chapter 6). If Paul intended to prevent women from leading, teaching, and serving, he would have done it in Ephesians.
But that wasn't the case. It was about 4 years later that Paul wrote to Timothy. At this point, the church at Ephesus was growing, but faced incredible pressure since Ephesus was one of the major Roman cities in the Mediterranean region. It was still a major city that had shrines to Diana, and significant polytheistic influences. Paul was concerned that women that had only began studying in the last few years could be easily corrupted and mislead the church.
Again, not a generalization that women should never teach or preach or lead, but merely that their situation was less than ideal for that expansion of women's rights into church leadership at that point.
In today's society, women are equally as educated as men (some would argue that many women are smarter and therefore more equipped to teach)