You posted to the wrong forum.

Privet (ligustrum) will make a dense hedge, but you'll be pruning 24/7 for the entire growing season. Not sure what variety I had, but it definitely did throw out suckers and would shoot out wayward branches four and five feet long. Within days of pruning there'd already be new growth starts. And if things couldn't get worse, beneath each shiny, waxy, hard to pick up out of the grass leaf there would be a million white flies that swirled up in clouds when disturbed. I only tolerated this "builder's special" because it had interesting contorted low branches. I'd prune it up at least twice a season to showcase the branching - wearing gloves and a face & nose mask. I can definitely understand your need for a screen - and fast. The major problem is as shrubs grow high, they also grow wide and giving up space becomes an issue. Upright evergreens would seem to be the ideal, though. Are you sure your Green Giants are goners? A pair of old pantyhose cut down the legs on the bias in a continuous rope does a fabulous job of tying up multiple leaders or a drooping snow bent branch. And browning can regenerate if the damage hasn't gone into the "dead zone" in the interior of the plant. Winter browning can be expected with most species and cultivars of Thuja. Some of the Taxus (yews) or junipers might come with higher recommendatons.

what is your personal goal .. immediate satisfaction.. or patience ... and your answer lies within ... if budget is irrelevant .. then fill it in ASAP ... why wait i dont see air flow as an issue.. since you want them to fill in completely anyway ... bottom line.. you are overthinking it... just tell us what you want.. right now .. dont think about it .... come on.. we are waiting.... ken

Privet will grow in part shade, it prefers full sun. Your plants may take some time to get started and may not get all that thick (full) in mostly shade.