עַל כֵּן אֵבָרִים שֶׁפִּלַּֽגְתָּ בָּֽנוּ וְרֽוּחַ וּנְשָׁמָה שֶׁנָּפַֽחְתָּ בְּאַפֵּֽינוּ וְלָשׁוֹן אֲשֶׁר שַֽׂמְתָּ בְּפִֽינוּ. הֵן הֵם יוֹדוּ וִיבָרְכוּ וִישַׁבְּחוּ וִיפָאֲרוּ וִירוֹמְמוּ וְיַעֲרִֽיצוּ וְיַקְדִּֽישׁוּ וְיַמְלִֽיכוּ אֶת שִׁמְךָ מַלְכֵּֽנוּ.
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Therefore, the limbs/organs which You apportioned in us, and the spirit and soul that You have breathed into our nostrils, and the tongue You have set in our mouth—behold, they will thank, bless, praise, glorify, exalt, revere, sanctify, and proclaim the sovereignty of Your Name, our King. |
Why were not more common verbs used such as the "organs that you created שֶׁבּראתָ in me" or "that you formed שֱׁיצרתָ in me" or "that you grew שֱׁגדלתָ in me?"

I would like to suggest that Chazal utilized the verb root פּ־ל־ג which most aptly describes how human organs are formed: they are hinting to the biological process of cellular division (mitosis) and differentiation.
Further support for this idea: consider the word for organ, א־ב־ר אֵבֶר.
One cell (א) divides into two cells (ב) in a continuous process, signified by the letter ר whose pronunciation indicates activity and movement (e.g., רץ run, רעשׁ noise).
Cellular mitosis was first witnessed in the modern era 200 years ago.
Therefore, the organs in which You continuously divide [cells] in us, and the spirit and soul that You have breathed into our nostrils, and the tongue You have set in our mouth—behold, they will thank, bless, praise, glorify, exalt, revere, sanctify, and proclaim the sovereignty of Your Name, our King

By Evilonan - Own work, CC BY-SA 4.0

1) RaDaK in Sefer HaSharishim for א־ב־ר cites examples of power, strong and anger. To split and duplicate onself certainly requires strength, but I am not certain how to further connect the definition.
Following is the two page entry from the Jastrow Dictionary for פּ־ל־ג :


(ב) נִשְׁמַת כָּל חַי
(ג) תְּבָרֵךְ אֶת שִׁמְךָ ה' אֱלֹקֵֽינוּ.
(ד) וְרֽוּחַ כָּל בָּשָׂר
(ה) תְּפָאֵר וּתְרוֹמֵם
(ו) זִכְרְךָ מַלְכֵּֽנוּ תָּמִיד.
(ז) מִן הָעוֹלָם וְעַד הָעוֹלָם
(ח) אַתָּה אֵל.
(ט) וּמִבַּלְעָדֶֽיךָ
(י) אֵין לָנוּ מֶֽלֶךְ גּוֹאֵל וּמוֹשִֽׁיעַ.
(יא) פּוֹדֶה וּמַצִּיל וּמְפַרְנֵס
(יב) וּמְרַחֵם
(יג) בְּכָל עֵת צָרָה וְצוּקָה.
(יד) אֵין לָנוּ מֶֽלֶךְ אֶלָּא אָֽתָּה.
(טו) אֱלֹקֵי הָרִאשׁוֹנִים וְהָאַחֲרוֹנִים.
(טז) אֱלֽוֹקַּ כָּל בְּרִיּוֹת.
(יז) אֲדוֹן כָּל תּוֹלָדוֹת.
(יח) הַמְּהֻלָּל בְּרֹב הַתִּשְׁבָּחוֹת.
(יט) הַמְנַהֵג עוֹלָמוֹ בְּחֶֽסֶד
(כ) וּבְרִיּוֹתָיו בְּרַחֲמִים.
(כא) וַה' לֹא יָנוּם וְלֹא יִישָׁן.
(כב) הַמְּעוֹרֵר יְשֵׁנִים.
(כג) וְהַמֵּקִיץ נִרְדָּמִים.
(כד) וְהַמֵּשִֽׂיחַ אִלְּמִים.
(כה) וְהַמַּתִּיר אֲסוּרִים.
(כו) וְהַסּוֹמֵךְ נוֹפְלִים.
(כז) וְהַזּוֹקֵף כְּפוּפִים
(כח) לְךָ לְבַדְּךָ אֲנַֽחְנוּ מוֹדִים.
(כט) אִלּוּ פִֽינוּ מָלֵא שִׁירָה
(ל) כַּיָּם.
(לא) וּלְשׁוֹנֵֽנוּ רִנָּה
(לב) כַּהֲמוֹן גַּלָּיו.
(לג) וְשִׂפְתוֹתֵֽינוּ שֶֽׁבַח
(לד) כְּמֶרְחֲבֵי רָקִֽיעַ.
(לה) וְעֵינֵֽינוּ מְאִירוֹת
(לו) כַּשֶּֽׁמֶשׁ וְכַיָּרֵֽחַ.
(לז) וְיָדֵֽינוּ פְרוּשׂוֹת
(לח) כְּנִשְׁרֵי שָׁמָֽיִם.
(לט) וְרַגְלֵֽינוּ קַלּוֹת כָּאַיָּלוֹת.
(מ) אֵין אֲנַֽחְנוּ מַסְפִּיקִים
(מא) לְהוֹדוֹת לְךָ ה' אֱלֹקֵֽינוּ
(מב) וֵאלֹקֵי אֲבוֹתֵֽינוּ.
(מג) וּלְבָרֵךְ אֶת שְׁמֶֽךָ
(מד) עַל אַחַת מֵאָֽלֶף
(מה) אֶֽלֶף אַלְפֵי אֲלָפִים
(מו) וְרִבֵּי רְבָבוֹת פְּעָמִים.
(מז) הַטּוֹבוֹת שֶׁעָשִֽׂיתָ
(מח) עִם אֲבוֹתֵֽינוּ וְעִמָּֽנוּ.
(מט) מִמִּצְרַֽיִם גְּאַלְתָּֽנוּ
(נ) ה' אֱלֹקֵֽינוּ.
(נא) וּמִבֵּית עֲבָדִים פְּדִיתָֽנוּ.
(נב) בְּרָעָב זַנְתָּֽנוּ.
(נג) וּבְשָׂבָע כִּלְכַּלְתָּֽנוּ.
(נד) מֵחֶֽרֶב הִצַּלְתָּֽנוּ.
(נה) וּמִדֶּֽבֶר מִלַּטְתָּֽנוּ.
(נו) וּמֵחֳלָיִם רָעִים וְנֶאֱמָנִים
(נז) דִּלִּיתָֽנוּ:
(נח) עַד הֵֽנָּה
(נט) עֲזָרֽוּנוּ רַחֲמֶֽיךָ.
(ס) וְלֹא עֲזָבֽוּנוּ חֲסָדֶֽיךָ.
(סא) וְאַל תִּטְּשֵֽׁנוּ ה' אֱלֹקֵֽינוּ.
(סב) לָנֶֽצַח:
(סג) עַל כֵּן
(סד) אֵבָרִים שֶׁפִּלַּֽגְתָּ בָּֽנוּ.
(סה) וְרֽוּחַ וּנְשָׁמָה
(סו) שֶׁנָּפַֽחְתָּ בְּאַפֵּֽינוּ.
(סז) וְלָשׁוֹן אֲשֶׁר שַֽׂמְתָּ בְּפִֽינוּ.
(סח) הֵן הֵם יוֹדוּ וִיבָרְכוּ
(סט) וִישַׁבְּחוּ וִיפָאֲרוּ וִירוֹמְמוּ
(ע) וְיַעֲרִֽיצוּ וְיַקְדִּֽישׁוּ וְיַמְלִֽיכוּ
(עא) אֶת שִׁמְךָ מַלְכֵּֽנוּ.
(עב) כִּי כָל פֶּה לְךָ יוֹדֶה.
(עג) וְכָל לָשׁוֹן
(עד) לְךָ תִשָּׁבַע.
(עה) וְכָל בֶּֽרֶךְ לְךָ תִכְרַע.
(עו) וְכָל קוֹמָה
(עז) לְפָנֶֽיךָ תִשְׁתַּחֲוֶה.
(עח) וְכָל לְבָבוֹת יִירָאֽוּךָ
(עט) וְכָל קֶֽרֶב וּכְלָיוֹת
(פ) יְזַמְּרוּ לִשְׁמֶֽךָ
(פא) כַּדָּבָר שֶׁכָּתוּב
(פב) כָּל עַצְמוֹתַי תֹּאמַֽרְנָה
(פג) ה' מִי כָמֽוֹךָ.
(פד) מַצִּיל עָנִי
(פה) מֵחָזָק מִמֶּֽנּוּ.
(פו) וְעָנִי וְאֶבְיוֹן
(פז) מִגּוֹזְלוֹ:
(פח) מִי יִדְמֶה לָּךְ
(פט) וּמִי יִשְׁוֶה לָּךְ
(צ) וּמִי יַעֲרָךְ לָךְ.
(צא) הָאֵל
(צב) הַגָּדוֹל הַגִּבּוֹר וְהַנּוֹרָא.
(צג) אֵל עֶלְיוֹן
(צד) קֹנֵה שָׁמַֽיִם וָאָֽרֶץ:
(צה) נְהַלֶּלְךָ וּנְשַׁבֵּחֲךָ
(צו) וּנְפָאֶרְךָ
(צז) וּנְבָרֵךְ אֶת שֵׁם קָדְשֶֽׁךָ:
(צח) כָּאָמוּר
(צט) לְדָוִד
(ק) בָּרְכִי נַפְשִׁי אֶת ה'.
(קא) וְכָל קְרָבַי
(קב) אֶת שֵׁם קָדְשׁוֹ:
(קג) הָאֵל
(קד) בְּתַעֲצֻמוֹת עֻזֶּֽךָ.
(קה) הַגָּדוֹל בִּכְבוֹד שְׁמֶֽךָ.
(קו) הַגִּבּוֹר לָנֶֽצַח
(קז) וְהַנּוֹרָא בְּנוֹרְאוֹתֶֽיךָ.
(2) The soul of every living thing
(3) shall bless Your Name, Adonoy, our God;
(4) and the spirit of all flesh
(5) shall glorify and exalt
(6) Your mention, our King, continually.
(7) From world to world,178See note 55 on תפילה למשה, page 145.
(8) You are Almighty;
(9) and besides You
(10) we have no king, redeemer, or deliverer,
(11) [who] liberates, rescues, maintains
(12) and is compassionate
(13) in all times of trouble and distress.
(14) We have no king, helper, supporter but You.
(15) God of the first and last [generations],
(16) God of all created things,
(17) Master of all begotten things,
(18) Who is extolled with a multitude of praises,
(19) Who conducts His world with kindliness
(20) and His creatures with compassion.
(21) and Hashem neither slumbers nor sleeps.
(22) He arouses those who sleep
(23) and awakens those who slumber,
(24) and He gives speech to the mute,
(25) and releases the imprisoned.
(26) He supports those who have fallen
(27) and straightens the bent.
(28) To You alone we give thanks.
(29) Even if our mouths were filled with song
(30) like the sea,180Just like the sea is filled with water.
(31) and our tongues with exultation
(32) like the roaring of its waves,
(33) and our lips with praise
(34) like the breadth of the firmament,
(35) and our eyes were radiant
(36) like the sun and the moon,
(37) and our hands outspread
(38) like [the] eagles of the sky,
(39) and our feet light as the deer—
(40) we would never sufficiently
(41) thank You, Hashem, our G-d
(42) and G-d of our fathers,
(43) and bless Your Name
(44) for even one thousandth
(45) of the billions
(46) and trillions
(47) of favors which You did
(48) for our fathers and for us:
(49) from Mitzrayim You redeemed us,
(50) Hashem, our G-d;
(51) from the house of bondage, You liberated us;
(52) in famine You nourished us,
(53) and in [times of] plenty, You fed us;
(54) from the sword You saved us
(55) and from pestilence You removed us,
(56) and from severe and lingering sicknesses,
(57) have You withdrawn us.
(58) Until now
(59) Your compassion has helped us
(60) and Your kindliness has not forsaken us.
(61) Do not abandon us, Hashem, our G-d,
(62) for eternity.
(63) Therefore,
(64) the limbs which You apportioned for us,
(65) and the spirit and soul
(66) that You have breathed into our nostrils,
(67) and the tongue You have set in our mouth—
(68) behold, they will thank, bless,
(69) praise, glorify, exalt
(70) revere, sanctify, and proclaim the sovereignty
(71) of Your Name, our King.
(72) For every mouth will thank You,
(73) and every tongue
(74) will swear allegiance to You,
(75) and every knee will bend to You,
(76) and all that stand up
(77) will prostrate themselves before You;
(78) all hearts will fear You,
(79) and all innards and kidneys
(80) will sing to Your Name
(81) as is written,
(82) “All my bones will say:
(83) Hashem, who is like unto You?
(84) You save the poor man
(85) from one stronger than he,
(86) and the poor and needy
(87) from one who would rob him.
(88) Who is like You?
(89) Who is equal to You?
(90) And who can be compared to You?
(91) The Almighty,
(92) the Great, the Powerful, the Awesome
(93) most high Almighty,
(94) Possessor of heaven and earth.
(95) We will extol You, we will praise You,
(96) we will glorify You,
(97) and bless Your Holy Name
(98) as it is said,
(99) “By Dovid:
(100) My soul, bless Hashem
(101) and all that is within me [bless]
(102) His Holy Name.
(103) [You are] Almighty
(104) in the power of Your might;
(105) Great in the honor of Your Name,
(106) Powerful forever
(107) and awesome through Your awesome deeds;