Not necessarily- Zodiac and Bulova used off the shelf JB Champion and Kreisler bracelets and just put their clasp on it. Hamilton used JB flexies which looked like drug store specials. You never know.
And if it’s too big for you, then it may be possible to remove a link. Gently pull two sections at the bottom to their extreme and hold it them there- look for any indication on the underside of a pin or groove system. The expandro’s you pull two apart and flip one link 90 degrees and twist is sideways- the pins that hold the link into the channels slide inversely making it clear the channel at the extremes and the link lifts out. Then you do the same to the next and reinsert it to put it back together.
It sounds strange- perhaps I’ll do a pictorial if I get the time in the next day day or so. Not saying this works the same way, but they usually had a way of resizing.