Sounds like a weak glow driver. Glow plugs don't burn out from such a short usage. A weak glow driver can start an engine and use what little capacity it has that one time and not glow enough to fire the engine later. Had it happen myself. Bring an extra cell with you or an extra driver. Also, the wall wart chargers for glow drivers often suck. I started charging them on my multi chemistry charger with such a massive improvement I can't even begin to describe. Old Nicads from an ancient (15yr old PLUS) cordless circular saw (9.6v) battery pack. Will take and hold over 2500mah on multi-chem charger; not more than 1000mah from wall wart charger.
I invite the OP to post pictures of the bottom (element end) of their glow plugs that "burned up".