We aren’t giving away Kyrou for nothing

Why is Kyrou in that deal? That's beyond absurd. Kyrou is a top prospect and Bozak can fill in DET need at 2C. Do a body swap and it benefits both teams. Biega for example. Bozak would be the Wings 2C and a guy they can flip at the deadline and Biega can be a taxi squad guy for STL. Also $5M off the books for the Blues.

tarasenko and steen will be on LTIR to start the season

Honestly that Faulk signing was a very bad decision

I don't think it would take Kyrou to move Bozak, but Detroit's not a charity. $5M is a lot of cap these days, even for a useful player. St. Louis would have to add a sweetener to entice the Wings.

The Red Wings are benefiting way more in this deal. That's pure stinginess.

You're mostly right, but consider it this way. The Wings are going to be bad this season. Adding Bozak and they're slightly better, but still bad. So the move doesn't really help them out that much. St. Louis would now have more cap to sign Vince Dunn, which is a huge benefit for them.
St. Louis needs to move cap, Wings don't desperately need a 2C right now. Wings hold all the leverage. Absolutely stingy, but why do favors for your opponents unless you can benefit greatly?

I see where you're coming from but this deal significantly helps the roster regardless of how you view this team playing this year. Being too stingy might lose respect GMs have for you around the league. Why would you want to deal with guys who are trying nothing but to obliterate you in trades?

In the Blues defense, the cap was projected to go up to ~88mil this year before the pandemic and if it did they would’ve easily resigned Petro. Faulk was probably brought on as ED fodder.

To be fair, doesn't every GM try to obliterate each other in trades? But you're right, reputation damage is a pretty valid concern. With COVID ongoing and 1/3 of the league over the cap, I think Yzerman could probably get away with being stingy without much loss to his reputation, at least for now.

In theory yes, you are hoping to fleece in every deal but do it with transparency. Not know from the get-go it's going to be an extremely lopsided affair. GM's don't have to make deals with you and if they do they instead will go out of their way to get back at you. Could even go into Offer-Sheet territory.

GMs being petty like that would only ruin their own reputation. Nobody wants to deal with someone who will act like a petulant child if they feel they lost a trade or didn't get what they wanted.
I think we're going to have to agree to disagree here. The Wings taking Bozak might cost them draft positions, which is far more important than being a little bit better at this stage of the rebuild. For that reason, Bozak is more of a negative to the team than a positive, at least in my opinion.