You are right
To use the first ability of Great Weapon Master feat, you only need to attack with a melee weapon. Dagger is a melee weapon and as long as you crit or reduce a creature to 0 hit points, you can make a melee weapon attack with your bonus action, including with otherdifferent weapon other than the one that procs the GWM.
I believe it is intentional, as you can see from this published article from Sage Advice:
When designing a feat with a narrow use, we consider adding at least one element that can benefit a character more broadly—a bit of mastery that your character brings from one situation to another. [...] shows that some of the character’s expertise with one type of thing—crossbows, in this case—transfers to other things.