The the Crux of the issue is the necromancer fans have to say exactly what they want to be able to do with their undeads in combat, exploration, and social situations and what they are willing to give up on.It's designable, but it's going to require some swings.
WOTC is (understandably) avoidant of summons, which makes sense. Even a single extra-NPC is a full new set of health to track, attack rolls to figure out, damage to deal out, etc. It can significantly increase the time a round takes.
But...it seems to me, embracing 4E-style minions could fix the problem. Let the necromancer either summon a swarm (single entity, just like we see in 5E) or a number of 1-hp minions that use the necromancer's spell-attack, with set-damage (no dice roll needed).
This is a steep-departure from typical 5E, but with a new leadership team in charge, maybe now is the time to try some out-of-left-field adjustments?
Thinking you can get 3+ turns is unreasonable.
Getting more than 2 extra turns in a combat is literally a 9th level spell (Time Stop).