After a single, probably not even bad, movie? Get a grip.
"Probably not even bad" isn't exactly high praise.
Gunn came out and publicly said the Supergirl script was so good he rushed Supergirl and made it the next movie after Superman, he talked the script up as if it was the best thing ever, he said he hired a great director. Now the movie is a massive bomb, it might not even reach $150 million worldwide and even the positive reviews mostly settle on "It's okay".
Gunn has made several questionable choices from the start. The DCU was supposed to be a reboot but he also decided to make Peacemaker part of the DCU but only in a kinda/sorta way with a reshot ending of season 1 removing the DCEU Justice League. That also means that The Suicide Squad is kinda/sorta in the DCU and he talked about possibly keeping Blue Beetle which was pretty confusing and muddled the message a bit, is it a reboot or is it not a reboot?
He also decided to make tv/streaming shows when this was arguably on of the choices that hurt the MCU the most despite some of the shows being really good but it still made catching up feel like homework because people assumed they had to watch the shows.
But okay, tv shows are now a thing but what's the strategy here? Green Lantern becomes a tv project while Clayface gets a movie? This feels completely counterintuitive, Green Lantern is a space cop, this screams epic space adventure on the big screen while Clayface is a really small character in the big picture and should be a prime candidate for a tv project if you do them.
A shared cinematic Universe should also have a at least a somewhat unified vision and feel coherent but so far it doesn't. Par too the marketing campaign of Superman was declaring that color, optimism and fun are back in superhero movies contrasting the project to the DCEU version. And they delivered, Superman was bright, colorful and optimistic but with Supergirl they immediately switch to moody, drunk mess on various brown planets and that despite the comic the story is based on being extremely colorful and gorgeous to look at. In the comic the colorful worlds are a contrast to the darker story and that would have worked for the movie too but nope, the went with "brown".
Not all movies can have the same look, Batman would by necessity be literally darker than Superman but Supergirl should match Superman more. The movie does have some bright scenes (mostly flashbacks) but the marketing certainly didn't show it, it was brown.
And then there's the elephant in the room that's Batman or the lack of Batman. The DCU needs Batman and it needs him sooner rather than later and Gunn wasn't willing to make the hard choice to either declare Matt Reeves Batman is part of the DCU wether Reeves likes it or not or end Reeves Batman and do the DCU version right away. It is simply insane to try and build a shared universe but let Reeves do his own Batman over there.
And if someone wants to argue that there's enough room for two Batmen ... maybe. But Gunn isn't doing that either, he is clearly delaying his Batman for the sake of Reeves.
And the bit we do know about The Brave and the Bold is that he hired the Flash director, writer and producer team to do it which doesn't fill me with confidence. Gunn has praised The Flash as a great movie and some people argued that was just corporate speak because he works for DC now but the fact that he hired the team responsible for that movie suggests he was genuine when he praised it.
He also said he generally wants to let creatives do their own thing as in "I hire great people and let them cook" but that's the wrong approach. You can do that when you do individual, unconnected movies but a shared universe has to be producer driven with directors getting marching orders and having their creative freedom limited.
I don't question Gunn's qualifications as the head of the DCU after one movie, I question it after a string of baffling decisions I cannot understand.
It's also worth remembering that James Gunn was just a producer on this, not the writer or director.
Just a producer is underselling his position. He is the writer's and director's boss, he had the authority to order changes at any point before, during and after production. And of course he's also the one who hired the writer and director in the first place.