You are correct about the Quran reference. Its context is confirmed by several hadith and classical tafsir that specifically state that the passage is referring to those "too young to have begun menstruation".
Then there is the issue of Muhammad (Islam's ultimate role model) marrying a a six year old and having sex with her when she was 9 (presumably he was waiting for the first period, the sign of adulthood in Islam).
So given all this it is clear that marrying and having sex with very young girls is considered acceptable in Islam.
However, that doesn't mean that many Muslims don't find it personally unacceptable. It's just another example of the conflict between people's own morality and the morality of the religion they follow.
To confirm the veracity of your account:
This is from Aisha herself:
"The Messenger of Allah married me when I was six, and consummated the marriage with me when I was nine, and I used to play with dolls."
أَخْبَرَنَا مُحَمَّدُ بْنُ آدَمَ، عَنْ عَبْدَةَ، عَنْ هِشَامٍ، عَنْ أَبِيهِ، عَنْ عَائِشَةَ، قَالَتْ تَزَوَّجَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم وَأَنَا بِنْتُ سِتٍّ وَدَخَلَ عَلَىَّ وَأَنَا بِنْتُ تِسْعِ سِنِينَ وَكُنْتُ أَلْعَبُ بِالْبَنَاتِ .
Grade:
Sahih (Darussalam)
Reference : Sunan an-Nasa'i 3378
In-book reference : Book 26, Hadith 183
English translation : Vol. 4, Book 26, Hadith 3380
Narrated `Aisha:
I used to play with the dolls in the presence of the Prophet, and my girl friends also used to play with me. When Allah's Messenger (ﷺ) used to enter (my dwelling place) they used to hide themselves, but the Prophet would call them to join and play with me. (The playing with the dolls and similar images is forbidden, but it was allowed for `Aisha at that time, as she was a little girl, not yet reached the age of puberty.) (Fath-ul-Bari page 143, Vol.13)
حَدَّثَنَا مُحَمَّدٌ، أَخْبَرَنَا أَبُو مُعَاوِيَةَ، حَدَّثَنَا هِشَامٌ، عَنْ أَبِيهِ، عَنْ عَائِشَةَ ـ رضى الله عنها ـ قَالَتْ كُنْتُ أَلْعَبُ بِالْبَنَاتِ عِنْدَ النَّبِيِّ صلى الله عليه وسلم وَكَانَ لِي صَوَاحِبُ يَلْعَبْنَ مَعِي، فَكَانَ رَسُولُ اللَّهِ صلى الله عليه وسلم إِذَا دَخَلَ يَتَقَمَّعْنَ مِنْهُ، فَيُسَرِّبُهُنَّ إِلَىَّ فَيَلْعَبْنَ مَعِي.
Reference : Sahih al-Bukhari 6130
In-book reference : Book 78, Hadith 157
USC-MSA web (English) reference : Vol. 8, Book 73, Hadith 151
This confirms without any doubt that Aisha was underage at the time she and the prophet consummated their marriage.
Should all Muslims follow the example of Prophet Muhammad?
I think that's the idea.