So as long as any travel ban comes with more aid (which seems likely), the info will be revealed as before.

Still each decision to reveal critical information would become a though political decision, where politicians would have to assess whether it would cause more harm than good. This could lead to delays and downplaying.

Also, no matter what politicians do, individuals also respond to incentives:
People who visited a blacklisted country might lie about it in order to travel, people from a blacklisted country could travel to a neighboring non-blacklisted country and then travel using forged documents. Once they travelled illegally, they may delay getting medical assistance when they get sick, lie about where they have been, lie about people they have been in contact with, etc.

It looks like an iterated prisoner's dilemma: if you start defecting, then other players will defect against you, yielding a worse outcome for everybody.