As far as I know, there are several problems with the census described in the Gospel of Luke, including that no empire-wide census occurred near the time of Jesus’s birth and that Roman censuses never required people to return to their ancestral birth places. If this is correct, why did seemingly no early Christians question these glaring problems?
Saying it's not at all accurate isn't quite right. The historic events described at the beginning of Luke are way off, but beyond that there's not a lot of independently verifiable information in Luke. It certainly matches up with the accounts of Jesus's life found in Matthew. Both Luke and Matthew are thought to draw on a couple of the same sources, including the canonical Gospel of Mark and the so called "Q Source." The Q Source is a lost document/documents, likely a collection of sayings associated with Jesus, that circulated in the first century, but has been lost to us. There is a large amount of information overlapping between Matthew and Luke, including some exact quotes and obviously the same stories. Some of that is from Mark (the earliest Gospel), and the rest is theoretically from this source.
In short, much of Luke is accurate to the records circulating in early Christianity because it uses those records as sources. There's no reason that Christians would suddenly doubt the historicity of their other documents because Luke got a few, relatively minor, historical details wrong.
Furthermore, Luke is not significantly less accurate (given his goal and his sources) than other writers in the ancient world. Roughly contemporary with Luke are many historians who were more than happy to fudge the details for their own biases or just lack information because they didn't have access to modern record keeping. For instance, Livy and Tacitus, two of the most valuable Roman historians, are both noticeably inaccurate in some cases. Livy in particular wrote many anachronisms (such as political offices and the powers associated with them) into his history. Both of them are also rife with bias and ahistorical stories about political figures to support that bias. Tacitus in particular is vitriolic toward Emperors he didn't like.
Luke is not so different. The greatest innacuracies in his account of the life of Jesus serve to justify beliefs about Christ already held by Christians. For instance, you mention the census. The census acts as a storytelling element to get Mary, Joseph, and Jesus in Bethlehem where Christians believe he was born.