Everything written in ancient Greek that is foundational to western literature, has already been translated, likely to a higher standard than most of the people trying to learn it.

Unless you wish to be part of an effort to further improve the quality of these translations, including to adjust them for the fact modern languages themselves are a moving target, just read those translations.

Modern Greek, on the other hand, is a living language with new art and culture coming from it. I may not be able to write "a cup of tea please" without misspelling tea, nor pronouncing it so badly they reply in English (as per my user profile), but this is infinitely more valuable than knowing if the ancient Greek character inviting people over for a meal is saying the people will eat the meal or be the meal.

Why would modern Greek be "irrelevant"? Millions of Greeks and Cypriots speak the language, along with minorities in other countries and a very large and well dispersed diaspora. Greece and Cyprus are major holiday destinations for northern Europeans. There are major writers such as Nikos Kazantzakis who have used modern Greek so there are cultural reasons to. Heck, I even like some modern Greek music, and am grateful for it, since it was one of the few things which kept me happy during lockdown.

You're right in saying Classical (inc. Koine) Greek is far more influential, but modern Greek is not "frankly irrelevant".