Nothing is ever "linked" to your MagicBand as such. Not dining, not park tickets, not on-property resort reservations.
All of those things are linked to your My Disney Experience profile. Think of the MagicBand like a key that "points" to your My Disney Experience profile. If the reservation is on the profile, the band will pull it up.
What are you hoping to do, exactly? When you tell them your name at the podium, they'll have your reservation right there regardless of whether you have your band or not. It won't really do anything.
Better.... There is no "one system" at WDW. Each Restaurant TRIES to maintain a local copy of your Reservation at the Restaurant level.
Why? Because the CENTRAL Database holding your Reservation Numbers goes down. OFTEN.
Your MB is indeed just a key into the Master backend Database. But a LOT of facilities have local copies of this data, to protect themselves from problems associated with Master Database failure. Your entry to your ROOM (MB), is stored at the local Hotel. Your DINING Data is stored at the Hotel, or Restaurant level. All to have a bit of backup WHEN the ancient Central Database Server pukes, or the Network goes down.
So - a suggestion: ALWAYS PRINT OUT YOUR RESERVATION NUMBERS. ALL of them. Carry this with you. PAPER.
When trouble hits? There is nothing like PAPER, with a Res Number on it, captured from MDE (capture the letterhead), to stop
"Our System is down" whining.
HOW?
1) In Chrome - go to MDE "My Reservations".
2) PRINT. If you can print to PDF? Great .... upload this to your Cell Phone.
CARRYING Hard copy Res Numbers, downloaded and printed from MDE, has saved our tail ends SEVERAL times. Paper is cheap, folds well, can be stuffed in a pocket, and is WONDERFUL when confusion erupts
.
All personal opinion - I'm not necessarily right, no one else is necessarily wrong.