You have to realize, the effects and costumes are expensive. There’s a reason why every CW show is shot in Vancouver. These shows are made on a cable network budget. WB/DC doesn’t have Disney money to throw at these and they don’t see a reason to, yet.
Now to answer the why in why those characters are gone:
Hawkman and Hawkwoman sucked as characters. They were easily, along with Vandal Savage, the weakest part of Legends of Tomorrow. There’s a reason the show took off in its second season, because it cut it’s weakest links.
Firestorm is only gone because Victor Garber is getting up there in age and I imagine wants to relax rather than work on a show for half a year. They could still have used the character as there have been many halves to Firestorm in the past, but Franz Drameh is an English native and probably didn’t like being away from home for so long.
Kid Flash’s departure was pretty much out of nowhere, because Keiynan Lonsdale basically wanted to take a personal break. He said he’s not done playing Kid Flash, but he wasn’t going to stick around for the current seasons.
Citizen Steel’s power is basically all CG work and thus was expensive so they barely used his power for more than a few seconds an episode anyway. Right now, they’re clearly doing some kind of story arc with him at the Time Bureau so we’ll see what happens.
Ray’s costume is probably one of the more expensive ones so they prefer he not use it very often, especially considering whenever he does use the suit to shrink or shoot things, it’s more CG work.
Killer Frost, again, more CG work, and honestly I prefer that she’s not as active, because I don’t think the Flash should have a “team”.
Amay’s Vixen left the show, but the actress came back for this season so who knows what they’re doing.
I think Cisco is better relegated to comedy, that’s when he’s at his best, and I agree about Constantine; his character is wasted on a show as comedic as Legends.
I’m actually amazed that Ralph’s powers look as good as they do, especially with the frequency in which he uses them.
Grodd is all CG, and there isn’t much they can do with a character that’s done what he’s done in the comics on a small budget; I mean the Gorilla City episode was bitterly disappointing.