Earl,
I cut off my garlic chives blossoms about the first week of July or so....before they go to seed. Then I usually get another full set of blossoms before the fall. And, yeah, they divide and conquer alright....hard to get rid of them, if you're a mind to do so.
Once it stops SNOWING I'm going out and spraying about half of what I've got out there to kill them off before winter. They're growing up in cracks all over the map.....which happens to be cracks in the concrete patio...or between slabs....or around the perimeters....or between the edge of the pool and the concrete patio....you get the idea. I'd be buried in them if I didn't kill some off.....but I'll have to be ruthless for next year, as my Dad wants to sell the house, and I don't think having all the bees buzzing around the patio will be a selling feature.
I can vouch for the flower vinegar. I've also made it with just the garlic vinegar stems or flowerbuds (scapes?). Yummmmmmm.