I think it's definitely fair to question why Nashville has been so keen to move Askarov, given their historical success at developing goaltenders. If this was the first time we were hearing Askarov rumors (after the Saros extension) then I would just chalk it up to wanting to find value in an elite goaltending prospect because they already have a prime-aged elite starting goaltender locked down long-term. But Trotz has been trying to move Askarov for over a year now, even before they had Saros locked down. There are a bunch of factors and any combination of them could be true:
1. They don't think Askarov has value as a backup given how highly he was drafted and how high his potential. It would be a wasted asset and it makes sense to trade him to shore up other positions given that Saros is locked in for their whole contending window (and beyond).
2. They don't think Askarov is very good or they know something that will limit his NHL outcome that we don't. Maybe it's a fundamentals problem that they can't get him to fix, maybe it's a persistent attitude problem. Maybe they don't like that he's a fiery guy instead of the classic cool calm and collected netminder. Whatever the case, they don't value him.
3. Trotz doesn't care for prospects that he has not personally drafted. We've seen Grier getting rid of a lot of DW/Jr.'s guys seemingly mostly because they aren't his guys and only partially because they aren't very good.
4. Nashville is attempting to be a contender (I don't think they are, but that's just my opinion) and Askarov is their top prospect; it makes sense that they would dangle him for immediate help, as the Sharks did every year we contended. This idea is hurt some by the fact that Trotz was trying to move Askarov+15 for 5 at the 2023 draft in the hopes of getting Will Smith, who is not a "win-now" asset, but I still think this factor plays a part in the situation.
I personally think it's a combo of all four of these factors, which is why I wouldn't be willing to give up anything more than one of Bystedt/Edstrom and a non-1st round pick. But to what degree #2 is true is the deciding factor for how smart it would be to trade for him.