There are two basic problems with natural burial: First, burial should be in consecrated ground. Secondly, the chief reason in Halacha for the placement of a tombstone above a grave is not in order to commemorate the deceased, but to warn Cohanim (those of the priestly caste) not to approach the grave, it being a Biblical prohibition for a Cohen to come into close proximity of a grave. If a grave was unmarked this would be problematic.
I am not aware of any shortage of spaces in Jewish cemeteries.
The London Beth Din