Why didn't Theoden 💀kill Grima Wormtongue?

[BalrogRingDestroyer]

Was he spared for the same reason that Gandalf suggested sparing Gollum? Gandalf even openly says that it would be just for Theoden to kill him and that to keep him around or send him away would, either way, likely be ill for Theoden.

If they'd killed Grima, Pippin wouldn't have gone messing with that stupid Plantir. Also, both Lotho Sackville Baggins and Saruman would still have lived.