Rav Schwab is quoted (in his son's sefer) as asking, why do you have to repeat shmoneh esrei if you skip Mashiv haru'ach umorid hageshem? Not every word in shmoneh esrei is essential, as shown in havineinu, and here, we're not even asking for rain, it's just a mention of the gevura of rain. He answers with the Gemara in Chulin brought by Rashi in Breishis 2:5. The passuk says "Now no tree of the field was yet on the earth, neither did any herb of the field yet grow, because the Lord God had not brought rain upon the earth, and there was no man to work the soil." Rashi says
כי לא המטיר: ומה טעם לא המטיר? לפי שאדם אין לעבוד את האדמה, ואין מכיר בטובתם של גשמים. וכשבא אדם וידע שהם צורך לעולם התפלל עליהם וירדו, וצמחו האילנות והדשאים:
Because… not caused it to rain: Why had it not rained? Because there was no man to work the soil, and no one recognized the benefit of rain. But when man came and understood that the rains were essential to the world, he prayed for them, and they fell, and the trees and the herbs sprouted. — [from Chul. 60b]
Rav Schwab says that we see that even though Hashem prepared the rain, He held it back for one reason: so that when Adam was created, he should recognize the benefit of rain and pray for it. We see that recognizing and praying for rain is not just "a tefilla." Hashem had arranged the creation of the world and the garden and of Adam so that the entire enterprise would be put into motion specifically through this tefilla. It is the father of all tefilla. If you skip it, then your tefilla is not a tefilla. (מכיר בטובתם של גשמים is precisely what is meant by "mazkirin gevuras geshamim" in mashiv haru'ach, and of course התפלל עליהם וירדו applies to v'sein tal umattar.)
He also points out that Adam, from the first moment of his creation, was called Nefesh m'malilah, a creature that speaks. What was the point of being able to speak if there was nobody to talk to? The answer is that there was- it enabled him to be mispallel.
In the beginning of Bava Kamma, the Gemara says "mav'eh zeh Adam." Why call Adam Mav'eh, the "one who supplicates?" The answer, especially in the context of Adam mu'ad l'olam, is that life is dangerous and tentative, and tefilla is the basic necessity of the human condition. In our mesorah, Man is "One who prays."
A reader, Avrohom, sent in a marvelous Ben Yehoyada, and I'll paste Avrohom's words here. By way of introduction, the "ne'elam" or "milui" of a letter refers to the unpronounced letters in the letter's spelled out name. Example- Aleph: the first letter of the letter's name is the letter aleph, and then you have the letters lamed and phei. So the ne'elam/millui comprises the letters lamed and phei.
A reader, Avrohom, sent in a marvelous Ben Yehoyada, and I'll paste Avrohom's words here. By way of introduction, the "ne'elam" or "milui" of a letter refers to the unpronounced letters in the letter's spelled out name. Example- Aleph: the first letter of the letter's name is the letter aleph, and then you have the letters lamed and phei. So the ne'elam/millui comprises the letters lamed and phei.
On your point that "In our mesorah, Man is "One who prays."", Rav Moshe Shapiro brings down a beautiful Ben Yehoyada that says that the neelam of Odom is Mispallel. (Aleph - the ne'elam is lamed and pey, the ne'elam of Dalet is lamed and Taf and the neelam of mem is mem). Rav Moshe Shapiro said that the ne'elam represents the essence of something, its fundamental identity. And therefore, an Odom is in essence a mispallel, that is what his existence rests on, or as you said "In our mesorah, Man is "One who prays."
The Ben Yehoyada is in Sanhedrin 110, as follows:
I want to show this visually. The name אדם has three letters: אלפ and דלת and ממ.
The primary letters, the pronounced letters, are אלפ, and דלת, and ממ. The "hidden" letters are אלפ and דלת and ממ.
The hidden letters, then, are לפ, and לת, and מ, equaling מתפלל.
And now, back to the original post.
שם (סנהדרין ק"י) זה ממונו של אדם שמעמידו על רגליו יובן בס"ד דאמרו בגמרא דאמר רשב"י יכולני לפטור העולם מדין הפלה על אשר אין מתפללים בכונה דלבם מבולבל מן צרות הגלות כשיכור, דכתיב שכרה ולא מיין והנה מי שיש לו ממון הרבה יהיה לו לב רחב תמיד ולא יתבלבל משום דבר כי הממון מרחיב הלב א"כ בעל הממון אינו יכול לפטור עצמו מדין תפלה אלא הוא מתפלל בטוב. והנה אם תמלא אותיות אדם כזה אל"ף דל"ת מ"ם תמצא אותיות המלוי הם אותיות מתפלל והמלוי הוא רגלים של אותיות הפשוט. ולז"א זה ממונו של אדם דייקא שמעמידו על רגליו שמעמידו בתפלה בשופי הרמוזה ברגליו של אדם דמלוי אדם הוא איתיות מתפלל דהאדם צריך להיות מתפלל תמיד ואינו יכול לפטור עצמו מדין תפלה
I want to show this visually. The name אדם has three letters: אלפ and דלת and ממ.
The primary letters, the pronounced letters, are אלפ, and דלת, and ממ. The "hidden" letters are אלפ and דלת and ממ.
The hidden letters, then, are לפ, and לת, and מ, equaling מתפלל.
And now, back to the original post.
A Bar Mitzva bachur, and a chassan, and others that enter a new phase in life, are given many brachos, and are graced with many gifts - gifts of natural ability, of their environment, of educational and financial advantage. All these gifts and brachos are wonderful, but if you aren't mispallel, without siyata dishmaya, these wonderful gifts will remain unused. טמן עצל ידו בצלחת גם אל פיהו לא ישיבנה. The rain was waiting, it was just hanging there ready to fall, but it would not fall until Adam Harishon was mispallel. It's like going into the bank where you have a big account, walking over to the teller, standing there like a golem looking at her, tipping your hat and walking out. The money is all there waiting for you. But if you don't ask, you're not going to get a penny. We even say Shma koleinu in Shmoneh Esrei, a tefilla asking that Hashem should hear our tefillos!
Yeshiva people, especially younger baalei kishron, have a tendency to belittle davening. Many fine talmidei chachamim stay up till three in the morning learning and miss tefilla betzibur- even zman tefilla, with the excuse of oneis sheina. I don't disagree with them. At that stage, Torah is far more important than tefilla, and it is not unreasonable to say that such a person is 100% Toraso umanuso and entitled to the hetter of Reb Shimon bar Yochai (Yerushalmi Shabbos 1:2 and Brachos 1:2), and not only pattur from davening, but even pattur from Krias Shman and Sukkah. The problem is that this becomes a habit that remains even when you didn't stay up till three. I used to have the bungalow next to Reb Moshe, and we would watch as he came outside to the porch at four in the morning to stand and say Tehillim. When he found out that we were watching him, he asked if he was disturbing our sleep, and we assured him that he was not. The point is that he could have been learning, or writing, or many other things that would appear to be better uses of time than saying Tehillim. But he did not think so, and you, too, should realize that tefilla and tehillim are precious both for what they are and for what they help you to achieve.