The 2,2,2 Bicycle is strained (13 kcal/mol for DABCO's hydrocarbon counter part). The interior angles of the carbon framework lock the nitrogen in more of an sp2 type hybridization, making the lone pair higher in p-character. You are right that it is a good leaving group and a good nucleophile. The argument of pKa for nucleophilicity is often cited, but it is not always a good model. Especially in amine systems. Herbert Mayr is a strong supporter of this theory and has shown repeatedly that pKa and nucleophilicities can differ dramatically. See ACIE, 2007, 46, 6176 and J. Phys. Org. Chem. 2010, DOI: 10.1002/poc.1707.
The alkyl groups of Et3N are out of the way, since it sits mostly as the pyrimidalized conformer (except in the transitions state between inversions, which it doesn't spend much time in of course)