I really wouldn't be surprised if it was replaced. Grandpa had the guitar for over 40 years. At least 25 of those years it got played every Sunday at church service, several times a week sometimes, and traveled all over mission trips. You can see the wear on the fretboard from his fingers it got played so much. It's quite likely if it did have the wooden crosspiece, it got broken and replaced.
We really don't care about value or originality. It's sentimental. Tonight I temporarily fixed the broken string, it was broken right at the ball end, so just shorted it slightly and reinstalled the ball end. After sitting 15 years, it was still almost in tune. Tuned it up, played what few chords I can remember from 20 years ago. Sounds pretty good to my untrained ears. I figure I'll put some new tuners and strings on it, just so we can play grandpa's guitar. Made grandma's night by playing it on the phone for her. Not sure it's the best guitar to learn on, but for now, it's what we have.