I am concerned how an overhang over the adjacent streets would look. I think it would be displeasing, and should be avoided at all costs.
It would not address limitations to the concourse, and deficiencies to washrooms, concessions, and, as you say, the press box.
It is unfair to make comparisons with the Avenir Centre (which is only four years old), but TSN loved the fact that broadcast facilities at the Avenir Centre were essentially "plug and play." They had broadcast spaces off the concourse on both sides of the arena at centre ice overlooking the rink surface. It made their job infinitely easier. They were very enthusiastic.
A complete demo and rebuild is necessary. I'm sure the trade centre could be repurchased (if the price is right). Using the entire block would also make it easier to include the Grand Parade as an ancillary outdoor events venue for major events and tournaments like the World Juniors and the Memorial Cup, including such things as a fan zone, big screen, stage and food trucks. Argyle between Duke and Carmichael could be pedestrianized to better integrate the arena with the Grand Parade.
Being able to integrate an outdoor events space into an adjacent arena can be very important. Again, the Avenir Centre does this to great effect. The plaza has a refrigerated ice surface, a stage, parking for food trucks and fire pits to keep warm.
The only way to accomplish all this is to demolish and rebuild.