If the provincial regs say a river is fly fishing only during a certain time, it is fly fishing only.
If you are referring to DFO's regs, they don't state the gear that you can and can't use because they don't regulate freshwater bodies of water.
"Fisheries and Oceans" Canada is responsible for tidal (salt) water species and MFLNRO is responsible for non-tidal (freshwater) species.
Of course, Salmon and Steelhead overlap. Salmon are anadromous, meaning they spend part of their life in saltwater before returning to freshwater to spawn. Steelhead too are anadromous, but while they are classified as a "salmonid", their biology differs. All else being equal, they have the same genes as rainbow trout, which is a freshwater species.
So for regulation purposes, salmon fall under DFO's umbrella, and Steelhead fall under MFLNRO's umbrella.
Regardless of the species you are fishing for and whether you may retain such species, MFLNRO will regulate the time(s) of the year the freshwater body of water is open, and dictate the fishing method you may use.
In terms of fish, DFO set the openings and closures for salmon, and MFLNRO set the openings and closures for steelhead.
If you have any system-specific regulation questions feel free to PM me.