Is it ? The language is evolving very fast. Compare English from the 50's to what is spoken now, take into account the major dialects around the world...
There are different kinds of stability. One is whether the same language is spoken informally. Another is whether people are able understand earlier texts. I'm pretty sure texts from the 20th and 19th century are commonly read in High School, TV shows from 1960s onward are watched, etc. Having the past ever-present makes it something current informal language relates to.
Many, many people speaking English also creates an incentive for stability at the same time it creates variation.
Or picked up pretty easily. P. G. Wodehouse and Arthur Conan Doyle use some anachronistic language but it’s still 99% understandable without a dictionary.
I don't think it has evolved much at all. New words, yes, and some words fell out of favor. But you can find a book written in the 50s or a movie from the 50s and you would have no trouble understanding it. I'm fact, without context, you likely wouldn't notice it was from 70 years ago.
I don't see much stability.