Bryan set the whole up so he could look like a hero and bring up the rescue whatever he wanted something from Kim.

Because he wanted a nice vacation in Paris where he could practice his deadly martial arts skills. If they took a ransom then he wouldn't have an excuse to take the trip.

How about "why would the traffickers pick up rich white American girls at the airport and risk provoking a huge international incident", instead of just going to Ukraine, Romania, or other impoverished Eastern European state where beautiful blonde girls will actually pay for the privilege of coming west to work as hookers? (If you ever visit Macau, you'll see swarms of them there.)
Admittedly, just plaing to the ignorance of the audience
Why would Liam not tell a soul in Paris his daughter had been kidnapped, leaving the trail to go cold for 16 hours while he flies over? He only succeeded in tracking her down by a ludicrous sequence of unlikely coincidences.
Possibly he's paranoid that someone he'd tell might leak it to the traffickers - his own contact in law enforcement was fully aware of and compliant with the operation, as it turns out
He'd know plenty of current and former CIA assets already in Paris or close who could have helped him and at least started investigating.
This part I could never buy - the movie actually shows us his friends, so we know he's got them. Are we supposed to believe a longtime CIA spook has only ONE Parisian contact? No one he trusts to at least suss out the situation for him?
(Though that raises a different issue - if his friend was a high-ranking cop, he'd be able to arrange Kim's release as the simplest solution and convinced the gangsters to go along)
But as was pointed out to me on this very board: this whole film makes the most sense as a divorced, non-custodial dad's revenge fantasy. That's it's emotional appeal. To satisfy that appeal, the distant dad needs to be the solitary hero, not coincidentally saving his family with the same skills that alienated his family.

He used Stuart's private plane. Probably easier for making his escape.
"Oh my God! You put a living room where the crack den used to be!"

The answer to "why" anything happens in this film has the answer "Because it created a reason for Liam to go ultraviolent on some sleazy evil foreigners".
How about "why would the traffickers pick up rich white American girls at the airport and risk provoking a huge international incident", instead of just going to Ukraine, Romania, or other impoverished Eastern European state where beautiful blonde girls will actually pay for the privilege of coming west to work as hookers? (If you ever visit Macau, you'll see swarms of them there.)
Why would Liam not tell a soul in Paris his daughter had been kidnapped, leaving the trail to go cold for 16 hours while he flies over? He only succeeded in tracking her down by a ludicrous sequence of unlikely coincidences.
He'd know plenty of current and former CIA assets already in Paris or close who could have helped him and at least started investigating.
There is no way on earth any French officials would sanction a group of white slavers kidnapping tourists in Paris to sell to Arabs. No matter how much bribes they paid. The cost to the country in terms of bad publicity alone would be immense, everyone involved would be jailed for a very long time.
How the hell did Liam just get on a plane and go home after killing literally dozens of people at the ritzy party, on the boat, etc? These weren't Albanian pimps, but rich and well connected businessmen and diplomats.

We do not negotiate with Terrorists, we do not pay ransoms to Terrorists.

He probably reckons they will take the money and still not deliver the girl.
Its that man again!!

What doesn't make much sense is why the kidnappers don't take steps to ascertain the ransom value of their victims and at least try to get ransom, in an amount much larger than their captives' sex slave value. They also have an incentive to return the captives upon payment since future ransom demands would lack credibility if they failed to return them upon payment. Here, the daughter would have been more than willing to volunteer to the kidnappers that her stepfather was very wealthy and willing to pay ransom.

He's probably smart enough to know that paying off ransoms has risks in itself. If the kidnapper is ruthless enough to engage in kidnapping he's probably immoral enough to go back on his word and stiff the family even after they pay.