That Allah has stated:
... فَانكِحُوا مَا طَابَ لَكُم مِّنَ النِّسَاءِ مَثْنَىٰ وَثُلَاثَ
وَرُبَاعَ ۖ فَإِنْ خِفْتُمْ أَلَّا تَعْدِلُوا فَوَاحِدَةً أَوْ مَا
مَلَكَتْ أَيْمَانُكُمْ ...
... Marry women of your choice, Two or three or four; but if ye fear
that ye shall not be able to deal justly (with them), then only one,
or (a captive) that your right hands possess ... [4:3]
Limits the common marriage (different from the temporary marriage) to four simultaneous wives. That you said Allah has not explicitly banned more than four wives doesn't seem to be correct, Allah has stated what is allowed and what he has not stated is, therefore, not permitted. To better understand this e.g. see the following verse:
أَلَمْ تَرَ أَنَّ اللَّهَ يَعْلَمُ مَا فِي السَّمَاوَاتِ وَمَا فِي
الْأَرْضِ مَا يَكُونُ مِن نَّجْوَىٰ ثَلَاثَةٍ إِلَّا هُوَ رَابِعُهُمْ
وَلَا خَمْسَةٍ إِلَّا هُوَ سَادِسُهُمْ وَلَا أَدْنَىٰ مِن ذَٰلِكَ
وَلَا أَكْثَرَ إِلَّا هُوَ مَعَهُمْ أَيْنَ مَا كَانُوا ثُمَّ
يُنَبِّئُهُم بِمَا عَمِلُوا يَوْمَ الْقِيَامَةِ إِنَّ اللَّهَ بِكُلِّ
شَيْءٍ عَلِيمٌ
Seest thou not that Allah doth know (all) that is in the heavens and
on earth? There is not a secret consultation between three, but He
makes the fourth among them, - Nor between five but He makes the
sixth,- nor between fewer nor more, but He is in their midst,
wheresoever they be: In the end will He tell them the truth of their
conduct, on the Day of Judgment. For Allah has full knowledge of all
things. [58:7]
And note how Allah has Himself extended His own stated example to include all the other possibilities as well.
That the holy prophet had more than 4 simultaneous wives is then according to another three verses revealed specifically about himself, not for the others, [33:50-52] (even according to Hadeeth from Imam Sadiq --peace be upon him-- what is stated in [33:52] doesn't mean something that was allowed for other believers became forbidden for the holy prophet --peace be upon him-- and he could have married women as much as he wish, ref [1]).
But then why common marriage being limited to four simultaneous wives? See the following Ahadeeth from [2]:
محمد بن يعقوب: عن علي بن إبراهيم، عن أبيه، عن ابن أبي عمير، عن جميل
بن دراج، عن زرارة، و محمد بن مسلم، عن أبي عبد الله (عليه السلام)، قال:
«إذا جمع الرجل أربعا فطلق إحداهن فلا يتزوج الخامسة حتى تنقضي عدة
المرأة التي طلق». و قال: «لا يجمع الرجل ماءه في خمس».
From Imam Sadiq --peace be upon him-- is quoted that if a man marries
with four, then divorce from one, he can not marry with a fifth until
the Eddeh time counted from his divorce is passed, since a man must
not have his sperms in five.
عن منصور بن حازم، عن أبي عبد الله (عليه السلام)، قال: «لا يحل لماء
الرجل أن يجري في أكثر من أربعة أرحام من الحرائر».
From Imam Sadiq --peace be upon him-- is quoted that it is not
permitted for a man's sperm to be collected in more than four free
wombs.
ابن بابويه، قال: حدثنا علي بن أحمد، قال: حدثنا محمد بن أبي عبد الله،
عن محمد بن إسماعيل، عن علي بن العباس، قال: حدثنا القاسم بن الربيع
الصحاف، عن محمد بن سنان، أن الرضا (عليه السلام) كتب إليه فيما كتب من
جواب مسائله: «علة تزويج الرجل أربع نسوة و يحرم أن تتزوج المرأة أكثر من
واحد، لأن الرجل إذا تزوج أربع نسوة كان الولد منسوبا إليه، و المرأة لو
كان لها زوجان أو أكثر من ذلك، لم يعرف الولد لمن هو، إذ هم مشتركون في
نكاحها، و في ذلك فساد الأنساب و المواريث و المعارف». قال محمد بن سنان:
و من علل النساء الحرائر و تحليل أربع نسوة لرجل واحد، لأنهن أكثر من
الرجال، فلما نظر- و الله أعلم- لقول الله عز و جل: فَانْكِحُوا ما طابَ
لَكُمْ مِنَ النِّساءِ مَثْنى وَ ثُلاثَ وَ رُباعَ فذلك تقدير قدره الله
تعالى ليتسع فيه الغني و الفقير فيتزوج الرجل على قدر طاقته، وسع ذلك في
ملك اليمين، و لم يجعل فيه حدا، لأنهن مال و جلب، فهو يسع أن يجمعوا من
الأموال، و علة تزويج العبد اثنتين لا أكثر، أنه نصف رجل حر في الطلاق و
النكاح، لا يملك نفسه، و لا مال له، إنما ينفق عليه مولاه، و ليكون ذلك
فرقا بينه و بين الحر، و ليكون أقل لاشتغاله عن خدمة مواليه.
From Imam Reza --peace be upon him-- is quoted that if a man marries
with four woman the child born is his, but if a woman marries with
more than one man the born child's father will not be known whose is,
since the men have share in a single marriage and in this is the
corruption of the off-spring, the heritage, and the Ma'aarif(?) [note
that maybe today one can identify the real father based on genetics
but 1. this identification is not free from mistakes, 2. no one can
guarantee this identification to be performed for every born if the
statistics rise up much, 3. the wisdom behind the rules of Allah are
not necessarily one or two, they can be very but not all of them known
to us or even understandable to us till the Judgment day and maybe
beyond!] And the reason why a man can have four wives is that men are
larger in number than woman [not only in those times due to wars and
etc men being killed or die at the work more than were killed and died
the women, but even from genetics if my mind could help, moreover that
men's death statistics has been always greater even now] and Allah
knows best and what Allah has stated that "Marry women of your choice,
Two or three or four" this is His decree that He has decreed for
simplicity of marriage for every reach and poor, so that every one
marries to the extent that his wealth would let him. ... [the rest of
Hadeeth is about non free women which is not the case for today life]
[1] البرهان فی تفسیر القرآن، جلد ۴، صفحات: ۴۷۸-۴۸۲
[2] البرهان فی تفسیر القرآن، جلد ۲، صفحات: ۱۸و۱۹