I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here.

Also some time ago I read about Grothendieck's "Denunciation of so-called “general” topology" with interesting comments also made here:

According to Winfried Scharlau's book, Grothendieck described his work in a letter to Jun-Ichi Yamashita as: "some altogether different foundations of 'topology', starting with the 'geometrical objects' or 'figures', rather than starting with a set of 'points' and some kind of notion of 'limit' or equivalently) 'neighbourhoods'. Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces...

So I am wondering what progress has been made here and in what directions. Does there currently exist an approach to the foundations of general topology that is not based on a notion of "points", in the spirit of Grothendieck's denunciation?