Bombardier courted Boeing, to be taken over.
Boeing made a huge mistake in not taking Bombardiers C-Series, because the C-Series was a great product to slot in to the Boeing series.
Unfortunately , it would also decimate the 737 market, since the 737 should have been put out to pasture 10 years ago, and I wonder if Boeing was still smarting from the MD takeover lessons learned.
As a result, Airbus stepped in - said thanks - and rebranded it the 220.
Airbus were not concerned about it decimating the A320 family, different mission profiles really.
Now, Boeing did have a purchase going on of Embraer, which they pulled out of in 2020, which is where your confusion lies.
Embraer spent a fortune reorganising to separate out the Military arm from Commercial. They are/were taking Boeing to arbitration as Boeing had penalties for pulling out...