See I don’t know if I believe the burning bridges part.
Again, I provided a real example of an offer sheet affecting a future trade going down. It's not just the GMs, but also the owners. David Poile was willing to deal with Philadelphia after the Weber offer sheet, but Nashville's owner apparently wasn't.
It's not just the GMs. We've heard stories over the years about owners holding long grudges and they get the final say over approving trades.
I’m almost sure nobody who has been offer sheeted hasn’t signed it. If someone has I certainly don’t remember it and it certainly hasn’t occurred since 1996 or 97 when I started following the NHL closely
Hard to find an archived article from October 2006, but Brian Gionta had to wait all summer to sign with the Devils since we were over the cap. At some point, he mentioned that his agent was given overtures about an offer sheet but Gionta declined.
Stuff like that would happen behind the scenes and usually wouldn't become public. A Gionta offer sheet at that point was plausible considering the Devils cap situation.
I'd imagine most GMs just wouldn't want to go through the effort of trying to sign a guy to an offer sheet if they knew the original team had plenty of cap room to match.