In terms of Leia in OBI-WAN, I don't believe there is a plothole or a continuity error, just a slight oddity in one scene of A NEW HOPE.
OBI-WAN KENOBI plays fair with the dialogue in A NEW HOPE. A NEW HOPE indicates that Ben only adopted his Tatooine alias after the Clone Wars. "'Obi-Wan... Kenobi. Now that's a name I've not heard in a long time."
Aboard the Death Star, Luke tells Leia, "I'm Luke Skywalker and I'm here to rescue you. I'm here with Ben Kenobi" and Leia reacts, "Ben Kenobi!" and immediately trusts Luke. This means that Leia must have met Ben during his civilian life to know him by that name.
This means OBI-WAN had space to show how Leia and Ben had met, although OBI-WAN now had to explain why Leia's holo message to Ben was, in light of OBI-WAN, a little odd:
Leia Organa's Holo Message in A NEW HOPE:
General Kenobi. Years ago, you served my father in the Clone Wars. Now he begs you to help him in his struggle against the Empire. I regret that I am unable to present my father's request to you in person, but my ship has fallen under attack and I'm afraid my mission to Alderaan has failed.
I've placed information vital to the survival of the Rebellion into the memory systems of this R2 unit. My father will know how to retrieve it. You must see this droid safely delivered to him on Alderaan. This is our most desperate hour.
Help me, Obi-Wan Kenobi. You're my only hope.
OBI-WAN KENOBI indicates that Ben's mission to rescue a young Leia was off the books. In contrast, Leia's message to Ben in A NEW HOPE was a formal request to recall a former soldier to the former Republic now Rebel Alliance. Therefore, she addressed him by rank ("General"). She referred to his last run of official active duty (his Clone War posting for Bail Organa).
She didn't refer to his off-the-books mission to rescue her because, if discovered, it would be apparent that Alderaan had been collaborating against the Jedi overtly after the Clone Wars when Alderaan was, at the time, attempting to maintain a fragile accord with the Empire ("Alderaan is peaceful, we have no weapons").
But doesn't Bail Organa, adoptive father of Leia, represent Alderaan? That doesn't appear to be the case in the non-LEGENDS, canon literature: the royal lineage was through Leia's adoptive mother, Breha Organa, which would allow some plausible deniability to declare that Bail had gone rogue, acted against his duty as a Senator in the Imperial Senate, and violated Alderaan's non-aggression pact with the Empire while leaving Alderaan untouched by his betrayal.
(In the end, it didn't matter; in A NEW HOPE, Tarkin says that Darth Sidious / Emperor Palpatine has dissolved the Imperial Senate and now Imperial soldiers have full authority to blow up Alderaan on a whim.)
The formal nature of Leia's message also made it clear: this wasn't seeking out Ben of Tatooine to do some discrete favour to maintain some secret arrangement with Dad; this was summoning General Obi-Wan Kenobi to resume active duty.
As for Leia's muted reaction to Ben's death in A NEW HOPE... life seems cheap and death seems constant in the STAR WARS universe. Leia earlier had a pretty muted reaction to her entire planet being destroyed. Luke has a pretty muted reaction to finding his farm burned and his uncle and aunt murdered; he actually gets sadder about Ben.
There's the possibility that Leia had a less than flattering remembrance of Ben. Ben staggered, blundered and stumbled in a rescue mission to save Leia's life, only had small bursts of actual Force ability that she witnessed, sent Leia away to survive while Ben parted ways to go on a suicide mission that he somehow survived, likely through a combination of luck, luck, luck and luck and possibly some brief burst of Force power.
She was grateful to him, but she probably didn't think that highly of him. My sense: Ben is the absolute last person that Leia would ever recruit for any dangerous mission; that if she turnned to old General Kenobi, she was down to her last potential agent.
In the context of Ben's not-great performance in OBI-WAN, Leia exclaiming "Ben Kenobi" implied that she was amazed that Ben actually made it to the Death Star when 10 years ago, he could barely make it to the spaceport.
Leia, seeing how upset Luke is, would not have wanted to tell her rescuer that her recollection of Ben was Ben being traumatized, helpless, terrified, defeated, unable to reliably access his Force-abilities, self-sacrificingly suicidal. She may have been wondering if Obi-Wan had finally achieved that death wish he had when they first met.
She may have been wondering if the frantic, frightened middle-aged man she met during OBI-WAN KENOBI had somehow gotten it together to become a great mentor adored by his pupil. She may have felt it best not to say anything until she ascertained who this Luke-person was and who this Ben-person had become and chosen to let Luke remember Ben Kenobi at his most self-assured and capable.
I have to note that Lucas had a peculiar attitude to tone in his movie. He wanted STAR WARS to be a lighthearted adventure story. Naturally, he set STAR WARS in a fascist dystopia.
Lucas wanted Luke and Leia to be fun, relatable characters. Naturally, he killed Luke's de-facto parents by incineration and blew up Leia's home planet while she stood by helplessly.
I think that's part of why ROGUE ONE had a relatively good response. It showed what life would actually be like under the Empire and in a rebellion. STAR WARS is the Saturday morning cartoon adventure version of that life.
I'm not sufficiently familiar with CLONE WARS to comment on Qui-Gon's continuity in that show.
I have to say, the reviews for OBI-WAN range from average to poor. The fanbase seems happy to have more Ewan McGregor, but the Serious Reviewers note that OBI-WAN is boxed in narratively: nothing that significant can happen to Obi-Wan, Darth Vader, Leia, Luke, Owen or Beru. A lot of viewers felt OBI-WAN could not rise above these restrictions.
And I think that's fair. But for me, I found a lot in seeing Obi-Wan reduced to the defeated Ben, horrified to discover that the supposedly-dead Anakin was now Darth Vader. The first scene where Ben encounters Vader and frantically runs was really gripping, as was Ben's eventual reunion with the Force.
One major criticism: reviewers seized on how Ben defeats Vader but doesn't kill him and walks away. This has been cited as OBI-WAN being restricted in what it can do with the characters even when it makes no logical sense for Ben to spare Vader. Later, Vader abandons the search for Ben; this was also cited as a forced character choice.
To me... Vader was defeated and I don't see Ben executing or assassinating anyone, ever; Jedi only kill in the heat of combat. Strategically, Darth Sidious had no love for Vader and would happily replace Vader with the next apprentice, so Ben was better off letting Vader stay where he was until Ben (or Luke) would be in a position to defeat both Vader and Sidious. As for Vader giving up on Ben, I would say that it showed how ultimately, Vader is a slave of Darth Sidious.
But it is an absolutely fair criticism to cite both as narrative problems.