These notions are based on flawed assumptions.
First, assuming the calendar has remained static for 2,000 years.
Christ’s birth was generally linked symbolically by Christians to the start of the new year.
At one time, that was mid-March, but later the end of December or early January.
The second flawed assumption is assuming Christians needed to appropriate pagan institutions.
The early history of Christianity was one of conscious rejection by pagans of their formerly pagan lives and customs.
Apart from Jesus’ Jewish followers, all Christians in the early church had at one time been pagans themselves.
Third flawed assumption is assuming the reverse is never the case. By way of example, Haitian ‘voodoo’ came about because Christian symbolism was appropriated to mask pagan rites.
One only has to witness the scenes of excess on our city streets this time of year to realize paganity has hijacked Christmas, not the other way round.
Last false assumption is assuming that if the exact date of a birthday is not known, it is somehow ‘inaccurate’ to celebrate it.
Less than 200 years ago few people knew when they were born, and even today, people may find themselves in that situation for a number of reasons.