Let the seed sprout and grow a little. Lay down boards to walk on (it spreads out your weight) and move them as needed to pull each buckthorn by hand.

Not much choice to do otherwise.

Looks to me as if the new berm has buried the lower trunks of at least one tree. It will kill the tree.

Unless you paid for significant weed-killing, which landscapers do by spraying weed-killer which often doesn't work well with shrubs like buckthorn, they just weed-whacked or mowed. Removing weeds is time-consuming.