I'm actually very confused about this:
The Hokage gets each individual mission report, debrief the ninja involved, decides which Jonin are available for missions that need them, etc. If not him, then someone needs to know which ninja are available for missions.
I understand the summaries can be wrong, and that's what eventually tipped off Shikamaru, but it seems a bit unimaginable that Asuma (and others) didn't see a report that said "12 total Jonin and special Jonin deaths" and then when figuring out who to assign a mission to either knew offhand (from the specific mission briefings) that those ninja were dead or kept naming names (more than 12!) only to find that, individually, each was dead.
Something like:
Asuma: This mission needs a special Jonin, who is available?
Clerk: Anko is available.
Asuma: Nah, I don't think she's a good fit for this mission, who else?
Clerk: No one else is available.
Asuma: Really? What about [name]?
Clerk: Dead.
Asuma: [Other names]?
Clerk: All dead.
Asuma: Wait, that's more than 12! What's going on?
This would have especially been the case earlier when it was more like 6 reported dead rather than 12 and Asuma would know who was currently on specific missions.
In a large nation I can understand the numbers being hidden, with easily fungible soldiers I can understand that, but Jonin (and special Jonin) are a very small core of known and named people and massive military assets whose deaths are immediately commented on and known.
There were 40 of them, total, now there's 10, but based on the reports people were thinking there were actually 28 still around. That's an 18 person discrepancy of some of your celebrity figures and major military assets. I don't understand how that would be missed or not identified much earlier as an error in the report.
Also, in terms of negotiations at AMITY, we should expect Rock to know when they've killed Jonin, they'd have a better picture of the war and our military ability than we did (though they wouldn't know how successful the collapse was) ....which then makes me wonder if Shikamaru also checked that the *positive* mission outcomes were also reported correctly. I'll assume he did because he's thorough, but having inflated good news of mission successes is easier to do and harder to suss out.