Buy a new door.
Unless there's something unusual about the door that's not evident from the description it literally isn't worth the time and effort to strip and refinish. Even if you price your time at $0.00 (which is fine, many do for home projects) there's a strong argument to be made that it's still not worth the costs involved, unless you ...
Stain remover worked fine for removing most of the stain and other upper lacquer/sealer layers.
A thin layer of veneer then was pressed on the door. All good without buying a new door.
Upload before/after photos, alongside details.
Update post after a year to report on the long-term effect.
Use stain remover, and chip a thin layer of wood, then apply a new stain. But I doubt that the pressed veneer is thick enough to allow for any thinning.
There's no fixing the existing veneer. Face veneers on plywood are very thin -- the internet says they're 1/30" on average, which means that half the time they're thinner than that. It might be a ...