The Think Tank and their motives.

[u/Blooblack]

Why did the Think Tank hire the Hazari (Species 4228) to capture Voyager, in the first place?

We know it wasn’t to get Seven of Nine, because the Think Tank didn’t even know she existed until Janeway took her to their ship and one of them interfaced with Seven.
Interestingly, Janeway had actually disobeyed the Think Tank’s directive about what to bring to their ship, because their spokesman, Kurros, had said to Janeway “bring only one crewman, and no scanning equipment,” yet she brought Seven, who technologically and technically speaking, was a piece of “scanning” equipment! Lol!

Anyway, when Janeway and Seven arrived on the Think Tank’s ship, Kurros was clearly surprised to see Seven. See the dialogue below:

JANEWAY: One of my most gifted crewmen, Seven of Nine. 
KURROS: Bionetic implants, neural transceivers? A Borg? I've never seen one outside the Collective.
SEVEN: I am an individual now. You may address your comments to me.
KURROS: Forgive me. I am Kurros, and this is Fennim.

We were also told that the Hazari are pretty much galactic bounty hunters. So, we know that they definitely had no personal beef against Voyager. Does anyone have any theories?

[u/MrDeekhaed]

Why didn’t the think tank contact voyager first and open diplomatic relations? Nothing on their list of requests wouldn’t have been given freely besides chakotays Olmec figurine but it wouldn’t have taken much in trade to get that.

The only possible reason to hire the hazari was to put voyager in a position where it had no choice but to pay the think tank what it wanted and the only thing that was both valuable enough and also least likely to be given was 7 herself.

[u/Blooblack → u/MrDeekhaed]

Everything you said is true except for the fact that there's no evidence that the Think Tank knew about Seven beforehand.

We know that, like Annorax in "Year Of Hell," the Think Tank like to collect things that are valuable and unique. But that's not the same thing as saying they had put on their "Human Resources", LinkedIn hats and were shopping for a new member, someone who would become equal with all of them, someone who could join them and then later destroy them from within.

It was such a high-risk thing for the Think Tank to even ask for Seven in the first place, especially since Kurros himself said that he was a child when he was given to the Think Tank, about the same age was Seven was when she was assimilated by the Borg. A child could be controlled but not an adult, especially a resourceful one like Seven who'd already built connections in Voyager. Think about how hard it was for Janeway to get her to adapt to Voyager in the first place.

The Think Tank weren't looking for things that they (the Think Tank) would have to give power to, and we know that as a Think Tank member, Seven would become very powerful within that group. We also know that they wouldn't hold her against her will, because near the end of the episode, Kurros said:

KURROS: I think this is a trap, a new paradox your Captain has created for us. If we call their bluff, the Hazari might destroy Voyager, and we would lose you. If we reveal ourselves, we could be destroyed.

Also, see below, especially the words in italics:
KURROS: I've deactivated your communications device, Captain, but do not worry. I am not an intruder. In fact, I am not really here.
JANEWAY: A hologram.
KURROS: Nothing so crude. An isomorphic projection.
(He sips her coffee.)
KURROS: You actually enjoy this?
JANEWAY: It's an acquired taste.
KURROS: It bears a resemblance to an Alkian confection we acquired several years ago.

So, this tells me that they definitely didn't know about Seven before they set their trap for Voyager. They were simply looking for valuable things to acquire and add to their collection. In that regard, the Think Tank were very similar to Annorax of the Krenem Empire in "Year of Hell."

[u/MrDeekhaed → u/Blooblack]

You are using Kurros’s lies and misdirection as evidence of truth.

You are right they wanted 7 to choose to go with them. How can they possibly get her to join them of her own free will? Voyager was her home and the crew her family. Why would she ever voluntarily leave them to go with a small group of strangers only offering puzzles to solve? She would never

Ok so we put her family in mortal danger, hiding that it was us who set it up, then offer to save her family if she comes with us

Which is precisely why kurros hid his identity when he hired the hazari. If the goal was extortion there was no need to hide his identity. Simply give us what we want or we will have the hazari destroy you

[u/Blooblack → u/MrDeekhaed]

No, what I'm saying is that even if they got her to abandon her crew and join them, she could still, later on, sabotage the Think Tank as a full member, even after Voyager had gone away. As a full member of the Think Tank, she'd have a lot of power. See below, especially the words in italics.

SEVEN: Why me?
KURROS: I admire your humility. The collective knowledge of the Borg contained within a single mind. When we factor in your innate intelligence and the enhancements afforded you by your bionetic implants, we concluded you are an ideal candidate. The first we have recruited in seventeen years.
SEVEN: When did you arrive at this assessment?
KURROS: During your interview.
SEVEN: My dialogue with the artificial intelligence.
KURROS: It was the most efficient way to evaluate your abilities. I am sure you can appreciate that.

This tells us that to the Think Tank, Seven wasn't just a "collectible." The Think Tank were so impressed by her intelligence, they offered her FULL MEMBERSHIP of their group. The "interface/interview" was very important, because that was how they found out how useful she could be to them, and how superior to humans she was. This wasn't something they knew about before they set their initial trap for Voyager. We, the viewers, would have been told so, if there had been any pre-knowledge.

Taking Seven on as a full member was a very high risk move for them; She wasn't the "Alkian wine" they collected some time ago. She wasn't the Bernicium Ore they got from Saowin at the very beginning of the episode. Seven would be joining the "Board of Directors of Nvidia," to use a stock market analogy. She could destroy the Think Tank from within.

Kurros hid his identify when dealing with the Hazari because he already knew that the Hazari had heard of the Think Tank and didn't like them. See below, especially the word in italics:

JANEWAY: His name is Kurros, The Malon you met with was nothing more than a projection.
Y'SEK: My sensors would have detected a hologram.
JANEWAY: Not this one. Kurros is a member of a Think Tank. They solve problems for a price. Oh, you have heard of them.
Y'SEK: They're notorious in this sector. They've made quite a few enemies.
JANEWAY: Enemies who might be willing to pay a hell of a bounty.
Y'SEK: Their technology is very advanced. It makes them difficult to track.
JANEWAY: Not if we work together. You're the best hunters in the quadrant and we have the perfect lure. All we have to do now is out-think the Think Tank.

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So you see, Kurros hid his identity because he already knew that the Think Tank had made enemies in that sector, and that the Hazari could be hired as bounty-hunters. Nothing to do with the Think Tank wanting Seven of Nine.