For example, π is pronounced "πι", or probably closed to "pee" in modern and in ancient Greek. It's never pronounced like "pie". Same with all letters that end with "i", for example "φ,χ,ψ" (pronounced as phee, chee, psee, never rhyming with pie). T (τ) was never pronounced as "ta-oo", either, not in ancient nor modern Greek.
There are differences between modern and ancient Greek of course. For example "β" (beta), originally pronounced more like it's now in English, only with a longer "e", while in modern Greek it's more like "vita")
Ancient Greek is needed to get a full Western education, for reading some of our foundational literature properly.
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.
I studied koine Greek with my dad. Today, I study Aristotle alongside half-a-dozen English translations (the latest, Adam Beresford's Ethics, is hilarious, "like Han Solo and Chewbacca, Achilles and Patrocles" in the notes; his Aristotle uses "Perhaps...but that's a bit off-topic").
None of the English translations is as convincing as knowing the original vocabulary. Many phrases and idioms are still obscure or debated. Why should the student not want to look behind the curtain?
Finally, there is something bracing about knowing the ancient grammar. Greek has features long-vanished from English.
You would separate students into those who never need to bother looking a bit into "foundational to western literature" and those handful who are on a PhD track. Eventually, nobody would grow up to be recruited into the latter.
I find ancient Greek not so helpful when it comes to etymologies. Some are helpful, but many are obscure or misleading. Climax comes from the word for a ladder apparently, and electron comes from the word for amber. There are stories behind both but they won't get you far. Any word beginning with psych- tends to relate to the mind, but the Greek means "soul".
You're right in saying Classical (inc. Koine) Greek is far more influential, but modern Greek is not "frankly irrelevant".