Early Christians were far more interested in Christ's death and resurrection than in his birthday. Emphasising December 25th as a festival was a later attempt to accommodate Pagans who, whilst wanting to be Christian, didn't want to loose their traditional festival. It was also a way of finding common ground between Christians and Pagans, by amalgamating the various contending beliefs under one umbrella - that of Sol Invictus (the Sun; this is why Sun-day is the Christian day of rest and of going to Church). Amalgamating various contending pagan beliefs in order to unify worship had been previously attempted by the Ptolomies and Romans with Serapis and Isis. Christianity doing it was never hidden by the Church Fathers - just forgotten by later generations.
Another reason to pick December 25th as the birth of Jesus was because the Annunciation (when the angel visited Mary and said she was going to be pregnant) was from an early stage believed to have occurred on or about the 25th of March. If Mary got pregnant about that time, then nine months later, when Jesus was born, would have been on or near the 25th of December.
The argument that Christmas couldn't possibly be the correct date because it is too cold for shepherds to be out with their flocks is curious, because surely the people who first celebrated Christmas as a Christian festival were living in countries where this would have been obvious to them. Either the date was chosen because it really was that date, or it was chosen as conveniently symbolic (the actual nativity date being unknown) and people knew that.