Cluster a categorical variable without breaking the existing categories
Without looking at the data, can't you just use your Domain Knowledge to create meaningful groups/attributes for the airlines data frame? Such as "low-cost carrier/premium", "foreign (non-USA) airline/US-Airline" , "long-distance carrier/domestic only", "founded in decade" ,  or whatever? With only 14 airlines you can assign appropriate values yourself, manually, without relying on some cryptic algorithm.

Besides, is your dataset identical to the dataset in the R package "nycflights13"?

If so, you could easily load it with library(nycflights13), after installing the package of course.