The reason is really quite simple: a census was ordered - as descendents of David, The Holy Family HAD to travel to Bethlehem from their home in Nazareth. Arriving late in the evening, in active Labor, they simply had no choice but to seek the nearest shelter, rather than travel around town looking for a distant relative that had room to put them up in a city that had tripled in population in just a few short weeks.
The Bible says “born in a stable and laid in a manger”. All the “scholars” in the world can’t alter the simple facts.
Why put quotes around something that's obviously not a quote? The word 'stable' never occurs, that part's only an inference made by later generations from the presence of the manger. The tale you recount is the popular version of the Christmas story, but is largely not supported by the actual text. Joseph would be a pretty lousy husband if he had really forced his wife to go on a long journey just as she was due to deliver. Nor for that matter would Rome have forced anyone to move around just to take a census. Just as with a modern day census the government cares where you live now (so they can tax you - some things never change), not where your family history is. Far more likely, the census was used as a convenient excuse to get out of Nazareth. After all, Mary was giving birth less than 9 months after the wedding, it's pretty obvious what the town gossips would conclude had happened.