Why doesn't Judaism put more of an emphasis on expressing our love for Hashem through music, song and poetry in a formal sense as in the Shul on days when musical instruments can be played. Or just with formal musical concerts and the like.
It does. Music has always been a huge part of the Jewish liturgical experience, across all cultures of Judaism around the world. Throughout the centuries, great poets have been composing piyutim and pizmonim -- and they continue still today -- in Hebrew, Aramaic, and their own vernaculars. Traditionally Jews have not been literate in music, for the most part, with the role of the cantor learned by talent and rote rather than theoretical learning and music theory. This is no longer true today, thankfully, but it might be one reason why most of the literally thousands of piyutim in our liturgical repertoire don't usually come with tunes attached.
That said, many tunes have been passed down traditionally for centuries -- not really that many centuries, but in the Ashkenazic tradition these melodies are called Misinai, from Sinai. They include many of the iconic pieces of High Holiday liturgy, like the great Aleinu and Kol Nidrei. Using the proper melodies for the liturgy in general is actually something halachah is actively concerned with. It is considered imperative to chant the liturgy or use tunes in the appropriate mode -- which does vary across cultures, but you should stick to the ones used by your own culture. For example, in Shabbat Shacharit, the proper nusach is to change modes from the Nishmat mode to the Ahavah Rabbah mode at "shevach" in El Adon. Do people do this? No, not really, but it is the supposed proper nusach.
Plenty of people put on concerts of Jewish music, including instrumental works and chazanut as well as folk songs and bakashot and z'mirot and you name it. And, of course, modern Jewish institutions train chazanim to use the correct melodies for all of the liturgy. We may not be allowed to use instruments halachically, but, first, PLENTY of people do so anyway, especially in the Reform communities of the world, and second, we still have our voices!
Also, while the center-right Orthodox establishment may not emphasize music as much as other groups and denominations, the Chasidim are way into music. The Baal Shem Tov taught that the niggun could elevate the soul spiritually, and his followers, even today, continue to use niggunim in this way. These niggunim are often written by Rebbes themselves, and it's important that they be sung precisely and with several customs -- for example, usually each strain of the niggun should be sung twice. (There is a reason, like always, but I don't remember what it is.) Each Chasidic group has its own melodies, some of which also become popular throughout Judaism.
Finally, Jews have an immensely rich musical tradition that varies across the world, across denominations, and across the liturgy and Tanach. Why is music education not stressed in traditional yeshivot? I don't know. I didn't go to them. Maybe it actually is stressed and I just wasn't told -- I know of several schools with strong cantorial programs. But even if they're mumbling, Jews are always singing. Z'mirot, nigunim, pizmonim, t'hilim, you name it, we're always singing. Maybe instrumental music we don't do as much since the Temple is no longer standing, but we haven't stopped singing since.