Sure, that would reduce the total ram by 1/8 ... But that would be a design choice to implement. Is ECC ram only 12.5% more expensive than non-ECC? If its higher, it may indeed be more advantageous to use non-ECC -if- a software compensation can be implemented.
In either case, extra memory accesses would be needed since checksums need to be loaded from memory. This would also make cache misses more frequent, since checksum data would evict non-checksum data from cache constantly. This would have a huge performance impact - most software contains a LOT of memory accesses.
However, it might be feasible to mitigate this in specific cases by having custom code in software that needs to be secure.
These checksums are typically done on blocks of payload data of course, not all memory content.
Most don't :)