"creating panic among many - particularly Indian passengers - who even chose to leave the aircraft"

Are they are getting off the aircraft because they believe the "fee" will be required of their employment imminently, and that their employer will not pay it, and this will lead to their visa getting cancelled before they could return to the United States?

IMHO, it seems like there's a good chance of confusion and delay when reentering the US in the middle of this kind of change. It would be better to avoid that, if possible. And in the case that your visa does get canceled, it would be easier to fight that from in the US, and if necessary, to wind down your US household from inside the US as well. Everything gets a lot harder if you have to do it from outside the country.

It seems for the moment they will only check for this new fee on entry at the borders. If the fee has been not paid entry will be denied from tonight.

Hence, if you stay in the country nothing will change. And they can wait until this gets played out in the courts, media, congress etc.

Or that, like the de minimus situation wrt post, there just is no process in place to pay the fee and you're left in a legal limbo, or worse.

It doesn’t seem very unclear - the president made up a policy it isn’t possible to comply with, since there is no way to actually pay the massive bribe, in addition to probably being illegal, but nonetheless CBP may start refusing entry to people in hours.