Consider two scenarios:
1. You sign Alex Pietrangelo, then trade two players, say Jake Allen and Jaden Schwartz, to get under the cap, potentially not getting as much in return.
2. You trade two players, say Jake Allen and Jaden Schwartz, to clear room under the cap (because everyone knows you want to sign Pietrangelo), then fail to sign Pietrangelo.
Which do you like less?
If at least one big-ticket player has to be traded to accommodate Pietrangelo's contract, seems to me you're better off waiting until after you sign him so you know you have to make that deal.
Once free agency starts, Pietrangelo is going to sign within the first hour, so you would have to have that deal already in place. Of course, if you create that space and Pietrangelo takes another offer, then you do have the flexibility to jump into the market yourself.
And I'm not sure what teams are lining up to acquire Steen. I think the Blues value him more than any other team does. Which is not to say they're making some kind of mistake. They know the value of him in the room and around the team better than anyone else.