My understanding is that burley is frequently (but not always) used as a relatively neutral base that helps create the melange that makes up the blend. Many of the burley blends are topped or cased heavily, which disguises the burley flavor but contributes to the mouth feel, body and finish.
I'm getting better at resisting TAD. Rather than order 4 or 5 tins of those suggested here, especially MSO, I went to the basement and found a can of Old Joe Krantz, circa 2007, which I'm enjoying while I type. Also grabbed a three year old tin of Paul Stanwell Jubilee, an aromatic composed of Virginias, burleys, orientals and dark fired Kentuckys.
I didn't think of myself as a burley fan but realized after reading your most entertaining and informative comments that my two all time favorites, and that of many others, is Esoterica's Stonehaven and Germain's Rich Dark Flake, both Virginia/burley blends.