Where the verses actually put the obligation
Read these as a standard rather than as a description of your Tuesday. They say what a community owes, and they say that knowing one another is a purpose, not an accident. Notice the third one in particular. Hearts being brought together is described as a favour Allah gave and an event that happened, not as the default setting of any room full of Muslims.
Qur’an and Sunnah
وَٱعۡتَصِمُواْ بِحَبۡلِ ٱللَّهِ جَمِيعࣰ ا وَلَا تَفَرَّقُواْۚ وَٱذۡكُرُواْ نِعۡمَتَ ٱللَّهِ عَلَيۡكُمۡ إِذۡ كُنتُمۡ أَعۡدَآءࣰ فَأَلَّفَ بَيۡنَ قُلُوبِكُمۡ فَأَصۡبَحۡتُم بِنِعۡمَتِهِۦٓ إِخۡوَٰنࣰ ا وَكُنتُمۡ عَلَىٰ شَفَا حُفۡرَةࣲ مِّنَ ٱلنَّارِ فَأَنقَذَكُم مِّنۡهَاۗ كَذَٰلِكَ يُبَيِّنُ ٱللَّهُ لَكُمۡ ءَايَٰتِهِۦ لَعَلَّكُمۡ تَهۡتَدُونَ
Hold fast to God’s rope all together; do not split into factions. Remember God’s favour to you: you were enemies and then He brought your hearts together and you became brothers by His grace; you were about to fall into a pit of Fire and He saved you from it- in this way God makes His revelations clear to you so that you may be rightly guided
Qur’an 3:103
إِنَّمَا ٱلۡمُؤۡمِنُونَ إِخۡوَةࣱ فَأَصۡلِحُواْ بَيۡنَ أَخَوَيۡكُمۡۚ وَٱتَّقُواْ ٱللَّهَ لَعَلَّكُمۡ تُرۡحَمُونَ
The believers are brothers, so make peace between your two brothers and be mindful of God, so that you may be given mercy
Qur’an 49:10
يَٰٓأَيُّهَا ٱلنَّاسُ إِنَّا خَلَقۡنَٰكُم مِّن ذَكَرࣲ وَأُنثَىٰ وَجَعَلۡنَٰكُمۡ شُعُوبࣰ ا وَقَبَآئِلَ لِتَعَارَفُوٓاْۚ إِنَّ أَكۡرَمَكُمۡ عِندَ ٱللَّهِ أَتۡقَىٰكُمۡۚ إِنَّ ٱللَّهَ عَلِيمٌ خَبِيرࣱ
People, We created you all from a single man and a single woman, and made you into races and tribes so that you should recognize one another. In God’s eyes, the most honoured of you are the ones most mindful of Him: God is all knowing, all aware
Qur’an 49:13
وَٱلۡمُؤۡمِنُونَ وَٱلۡمُؤۡمِنَٰتُ بَعۡضُهُمۡ أَوۡلِيَآءُ بَعۡضࣲۚ يَأۡمُرُونَ بِٱلۡمَعۡرُوفِ وَيَنۡهَوۡنَ عَنِ ٱلۡمُنكَرِ وَيُقِيمُونَ ٱلصَّلَوٰةَ وَيُؤۡتُونَ ٱلزَّكَوٰةَ وَيُطِيعُونَ ٱللَّهَ وَرَسُولَهُۥٓۚ أُوْلَٰٓئِكَ سَيَرۡحَمُهُمُ ٱللَّهُۗ إِنَّ ٱللَّهَ عَزِيزٌ حَكِيمࣱ
The believers, both men and women, support each other; they order what is right and forbid what is wrong; they keep up the prayer and pay the prescribed alms; they obey God and His Messenger. God will give His mercy to such people: God is almighty and wise
Qur’an 9:71
Translation: M.A.S. Abdel Haleem, The Qur'an, Oxford University Press, 2004.