Not even close to being true. When Alaric challenged Seth Ward after the turn that stopped Hammerfall he was responsible for overseeing the recovery & integration of the Wolves into their new territory. He combed the the planets to pick up every single military asset and for political purposes stopped some building projects like the Clan government building on Gienah to keep the Clan's senior leaderships from understanding he built three whole galaxies to be used in the second Lyran conflict along with the logistical support that would be needed to keep the ammo bins stocked.
All of what you said are examples of Alaric dealing with the war footing. A focus on producing bombs and bullets and warriors for his crusade. There's not much showing he actually has a clue how a basic spheroid works, thinks, or feels. KSD certainly didn't give him a good ground level view or the universe.
Also, worth noting that the early stuff about the Empire painted a pretty peachy image of the Wolf conquest, making it look like a breeze. EA shows that wasn't actually the case. There were a bunch of cracks painted over in earlier sources.
Which is Chance's mis-perception, Anastasia has not stopped being Clan nor are those from the Clans in the SLDF- Darren competed for a Clan Wolf bloodname, not a Star League bloodname for example.
Per trial of birthright, Anastasia would still be a wolf, same as Darren and anyone else, but she wouldn't be part of the touman. So I'm not clear if she would count as part of the clan council anymore. If she isn't, she'd be ineligible to become Khan. There's the possible counter-example of Natasha having been gone for decades and returning, but the Wolf's Dragoons were, fundamentally, a Clan Wolf unit, obedient to Kerlin Ward and Ulric Kerensky, following their orders to the hilt.
Mostly though, politics, not letter of law. Election to Khan is, at its heart, a popularity contest with a dash of murder. In ToB we see Chance, Anastasia, Noritomo, and by extension, everyone else drawing a distinction between the SLDF and Clan Wolf. Wolves are dismissive and arrogant of the SLDF, for they are the ilClan, etc., etc.
At some point, either when Alaric grows up and steps down as Wolf Khan or dies via bullet, heart attack, Seventh Kommando, Fidelis, Sephanie Chistu, or choking on a Big MAC (unit or burger, your choice), there's going to be a popularity contest. Anastasia's position as Commanding General of the SLDF is going to completely gut her chance of ever winning the votes to become Khan.