The no trade clause is the issue insofar as he can't be traded to any of the 10 teams on his no trade clause list without him expressly waiving that clause.
You seem to think it's not an obstacle at all, because if the Sens try to move him to one of those teams, and he refuses, the Sens can say "either waive it or we'll send you to the AHL".
If sending Zaitsev to the AHL like you want the Sens to threaten to do would solve the problem they're trying to solve (whatever that may be), they would simply do that first without trading him along with a high pick or good prospect. Because why would they pay one of those to move him when simply waiving him would solve the issue the Sens are trying to solve by trading him?
Waiving him and sending him to the AHL, therefore, doesn't solve the problem. Therefore, threatening to do so if he doesn't waive his NTC would be an empty threat, because ultimately they would still need to trade him because waiving him didn't solve the problem.
That's why they need to attach the pick and/or prospect to get the deal done.
So yes, the value required to move him is an obstacle, but you also need to work around his NTC to find a team willing to take him, and that he's willing to go to.