Thanx to Humphreys and Koen for providing, that rank 1,2 is "geometrically unsatisfying" in the sense of Tits buildings and suggesting the notion of split BN-pairs as a tightning. But looking at the suggested source (and others) I'm less convinced this helps answering the question....(?)

Split BN-Pairs of rank 2 (or "Moufang polygons"?) were classified (as you said) by Fong-Seitz and the result is (online e.g. in ">this secondary PDF), that the're basically:

$$A_2,\;,B_2,\;^2A_4,\; ^2A_5, \; G_2,\;^3D_4,\;^2F_4$$

Split BN-Pairs of rank 1 (or Moufang sets) were shortly before classified by Hering, Kantor, Seitz (online as PDF here) as close to (does the occasional double-occurence to above worry me?):

$$A_1,\; ^2B_2, \; ^2A_2,\;^2G_2,\;\text{or sharply 2-transitive}$$

However none of these appears to me a sporadic simple group? It still seems to me, that the prominent $B,N$-construction (e.g. Griess himself) of the monster is rather a weakening. Keeping things provable might be the reason for the complicated definition of a weak BN-Pair in Aschbachers Book (s.a.)- any clues on this? I couldn't even convince myself this IS the notion we look for?

...BUT I found another source: Parker e.g. uses weak BN-pairs of rank 2 (!) to distinguish odd characteristic Lie groups paper e.g. here. And I found (appearently quite different) weak BN-Pairs of odd order p...