Importance of religion: Why Islam?

Disclaimer:
Please view the accumulated links to have a clear understanding of the concept. Jazāk Allāhu Khayran.
Key Question(s):
- Why do we need Islam?
- Is there any proof that Islam is the correct religion?
Reference Quran:
إِنَّ ٱلدِّينَ عِندَ ٱللَّهِ ٱلْإِسْلَـٰمُ ۗ وَمَا ٱخْتَلَفَ ٱلَّذِينَ أُوتُوا۟ ٱلْكِتَـٰبَ إِلَّا مِنۢ بَعْدِ مَا جَآءَهُمُ ٱلْعِلْمُ بَغْيًۢا بَيْنَهُمْ ۗ وَمَن يَكْفُرْ بِـَٔايَـٰتِ ٱللَّهِ فَإِنَّ ٱللَّهَ سَرِيعُ ٱلْحِسَابِ
Indeed, the religion in the sight of Allāh is Islām. And those who were given the Scripture did not differ except after knowledge had come to them — out of jealous animosity between themselves. And whoever disbelieves in the verses of Allāh, then indeed, Allāh is swift in [taking] account.
[Surah 3, verse 19]
Tafsir (Ibn Kathir):
Reference Hadith(s):
Relevant Hadith:
لَيَأْتِيَنَّ عَلَى أُمَّتِي مَا أَتَى عَلَى بَنِي إِسْرَائِيلَ حَذْوَ النَّعْلِ بِالنَّعْلِ حَتَّى إِنْ كَانَ مِنْهُمْ مَنْ أَتَى أُمَّهُ عَلاَنِيَةً لَكَانَ فِي أُمَّتِي مَنْ يَصْنَعُ ذَلِكَ وَإِنَّ بَنِي إِسْرَائِيلَ تَفَرَّقَتْ عَلَى ثِنْتَيْنِ وَسَبْعِينَ مِلَّةً وَتَفْتَرِقُ أُمَّتِي عَلَى ثَلاَثٍ وَسَبْعِينَ مِلَّةً كُلُّهُمْ فِي النَّارِ إِلاَّ مِلَّةً وَاحِدَةً قَالُوا وَمَنْ هِيَ يَا رَسُولَ اللَّهِ قَالَ مَا أَنَا عَلَيْهِ وَأَصْحَابِي
Narrated ‘Abdullah bin ‘Amr: that the Messenger of Allah (ﷺ) said: “What befell the children of Isra’il will befall my Ummah, step by step, such that if there was one who had intercourse with his mother in the open, then there would be someone from my Ummah who would do that. Indeed the children of Isra’il split into seventy-two sects, and my Ummah will split into seventy-three sects. All of them are in the Fire Except one sect.” He said: “And which is it O Messenger of Allah?” He said: “What I am upon and my Companions.”
[Jami` at-Tirmidhi 2641: Book 40, Hadith 36]