I haven't thought of a three-team trade like you have here, but I've definitely thought about Chychryn. In theory, the Blues could acquire assets (say, two first-round picks) in trades for Tarasenko and O'Reilly, then flip them to Arizona for Chyrchrun. The Coyotes get their price. The Blues don't have to wait on a draft pick to develop in order to rebuild.
There are two issues that have been brought to me. One, you need to make sure that Chychrun is an all-around No. 1 defenseman. He's always driven play with the Coyotes, and produced offensively, but if you're making a bet this large, the evaluation needs to be correct. Think about Chicago and Seth Jones. In his first season in Chicago, it was a common refrain around the league that he is simply not a No. 1 defenseman. If Chychrun is another No. 2 or 3, and you add him to the group of Parayko, Krug and Faulk, the problem could get worse instead of better.
Two, Chychrun is only signed for two more seasons at $4.6 million. Ideally, you'd want to contend during the seasons in which your top players are underpaid. Will the Blues be that during those two seasons? And will St. Louis be able to re-sign him at the end of his contract, when he'll be a UFA? Worst case scenario, it's a large price to pay for a two-year rental on a non-contending team.
Then there's the whole deal of working out a trade with Bill Armstrong and his deep knowledge of the Blues organization.