Indeed. I have imagined that Russia could assign cruisers from the other fleets. But that would also mean depriving a cruiser from the other fleets. In the context of the high tensions with NATO today, it may not be too helpful to the other fleets.Russia still has four Soviet-era heavies: Pyotr Velikiy, Admiral Nakhimov, Varyag and Marshal Ustinov. Either of the latter two could be reassigned to the Black Sea Fleet if Russia still wants to maintain that kind of presence.
That said, even if Russia were to assign e.g. Marshal Ustinov as the new flagship of the Black Sea Fleet, it would not have the same symbolism as Moskva did prior to her sinking. Everyone would know, or at least think (which is much the same for the purposes of prestige) that this is just another big Soviet rustbucket that can easily be sunk by anyone who cares to take a shot at it. Indeed, that's going to be the image for all of the Soviet-era heavies going forward.
I'm not suggesting that Russia should retire these ships because they no longer strike fear into the hearts of their enemies. Obviously such decisions should be made according to a clear-eyed assessment of capabilities, vulnerabilities, costs and alternatives. But strictly in terms of the prestige of the Russian Navy, the loss of Moskva under these circumstances is a massive blow that the service will likely not recover from for a generation.
Nevertheless, I do agree with your point that the sinking of Moskva have dealt a heavy blow to the once-feared image of those large Soviet-era cruisers in the Russian Navy. Now, instead of deterrence, NATO Naval planners might be emboldened to think that these Russian large cruisers are tinderboxes, which could be sunk with relative ease. Regardless if its actually true or not. It might weaken the overall non-nuclear deterrence effect of the Russian Navy in the near future.
The Russian Navy is in dire need of fleet modernization and replenishment of major surface vessels. Their sole aircraft carrier is out of action for a couple of years to come. They have just lost a Slava class cruiser. They have only 1 Kirov class operational, with another spending decades in the shipyard undergoing refit. Worse of all, Russia has no modern air defence destroyer or cruiser in 2022. If Russia is gonna be prideful and continue waiting for its domestic shipbuilders to deliver them, they are gonna have to wait decades more and spend billions of rubles more. Let's see if they can finally wake up from their erroneously low opinion of Chinese warships.