Are there any drawbacks to doing this?
There are softwares that are not going to be able to read BIGTIFF. It is likely that most geospatial/GIS software that consumes raster data can now handle BIGTIFF, but others might not have been updated or make assumptions about maximum sizes. BIGTIFF support did and does require some software implementation.
If you want your data a little more future-proof, if you use very old software (5+ years old), or you push your TIFFs back and forth between different disciplines (graphic design, GIS, photo, engineering), don't use BIGTIFF if you don't need it.
Why isn't BIGTIFF on by default, especially nowadays where >4 GB GeoTIFF files are quite common?
Ten years ago, when BIGTIFF was first implemented, they weren't so common. BIGTIFF's implementation was recognition that this was soon going to be a common problem.
GDAL tries to be smart about its use of BIGTIFF, but things can still fall over, especially when merging, etc. The advice from MappaGnosis is really good -- use VRTs for merging instead of writing huge TIFFs.
I'm still slightly disappointed that my proposed name of BFT did not win over the BIGTIFF one.