I'm not a medical doctor and not necessarily current on the latest research. However, there is quite a lot of evidence that general anesthesia is fairly likely to cause actual brain damage in infants and young children [1,2] with significant consequences in later life and there are (less significant, but still non-trivial) issues with various methods of local anesthesia. As far as I know, there's a lot of unsolved issues in the field of infant anesthesiology and the general trend is to err on the side of caution.

[1] http://www.webmd.com/parenting/baby/news/20120820/anesthesia...

[2] https://www.fda.gov/ForConsumers/ConsumerUpdates/ucm364078.h...