Why are men and women not separated during tawaf, sa'i, and prayer in the Grand Mosque, following the example of the Messenger's Mosque (peace be upon him), despite the severe crowding that leads to men touching women and vice versa, and who bears the sin of this intermingling?

Mixing between men and women is not permissible if there is a fear of fitna (strife/temptation) accompanying it, and this applies to the Sacred Mosque even more so. What is currently happening with the mixing around the Kaaba is impermissible. A Muslim should avoid mixing as much as possible, but if it happens unintentionally, there is no harm.