You might check out: Straight Dope Mailbag: Why do we have leap years? written by an illustrious Straight Dope Staffer.
The question is what you mean by accuracy.
Accuracy depends on what you think is important.
The solar year is “accurate” in the sense that, with minor correction every so many years, it maintains the days of the year in close correspondance with the position of the earth in its orbit, say within a day or so. Thus, March 21 and Sept 21 are always the equinox, to within a day or so error; in the northern hemisphere, August is always midsummer and Christmas is always in winter, etc. The calendar corresponds to the solar year and hence to the seasons.
The lunar calendar is accurate in the sense that it measures the passage of time by the phases of the moon. The phases of the moon come at the same time each month (for instance, full moon is exactly mid-month, etc.)
If you think the phases of the moon are important, then you want a lunar calendar. You note that the solar calendar is flawed, because the full moon falls on a different day each month. If you think the seasons are important, then you want a solar calendar, and you think the lunar calendar is flawed because the holidays aren’t seasonal.
Islam uses a pure lunar calendar, and the days of the year do not correspond to the seasons. One year, Ramadan is in the summer; a few years later, it is in the spring; etc. The lunar year falls short of the solar year by around 11 days (someone will correct me if I’ve misremembered.) The early Muslims were not farmers, and it was not important that holidays fall at certain seasonal times.
The Jewish calendar follows a lunar calendar with adjustments; every so often a full month is added, so that the holidays come out in the right season. The early Israelites were farmers and shepherds, and the Jewish holidays correspond to harvest and planting time in ancient Israel. On the other hand, the new moon was important, and so a calendar was developed that was mostly lunar but adjusted to roughly reflect the seasonal year.