What if some of these “hostages” sympathize with HAMAS?
That does not really matter at all if they are "Americans".
The lesson to teach our adversaries is "Don't F*** around with Americans or you and your family and all your friends will die". And to make that lesson stick, we will need to actually go all the way at least once or twice with a practical demonstration. This is the same problem facing the IDF. We "don't believe that is who or what we are".
Times and great trauma can change deeply held beliefs in some people. Not all people.
Nobody in the Middle East takes Russian hostages. There are good reasons why that is so. We must learn from our adversaries; they are our greatest teachers.
We can quietly hang the treasonous "hostages" afterwards, if they aren't killed in the retaliatory strikes first. Or if HAMAS does not kill them first.
It is not about saving the hostages. For all practical purposes they are already dead. It is about making adversaries fearful of planning to take hostages as a tactical advantage.
When tactics fail, the enemy will stop using them. Or they will keep using them. In the latter case, the enemy must be pressed into extermination. Ineffective enemy tactics make that much easier.
War is an incredibly ugly business that makes permanent changes in the world. Best to win quickly and decisively with ruthless and unrestrained ferocity so that it is over as soon as possible. Then do what we can to restore the souls of the people who had to do the fighting.
We are not truly prepared for war right now.