Based on the suggestions you’ve made:
If the General received a Good after Jarek’s death, that would be heavy info against Jarek, wouldn’t it? That would have been very strong IMO.
And in general, if the primary decision the General’s information is based off of is the previous day’s execution, that would frequently end up similar to an alignment cop (which perhaps is fine if the script is built around it - Undertaker and Oracle do exist, after all - but I would say that when building the script the way general is being run would have to be kept in mind)