I cannot see any advantage in a multi-national HQs. Why are they required to run multi-national formations, divisions or Corps?

If I am UK brigade, I am quite happy to have Dutch Tank company attached. At most I may want a Dutch LO, but that's it. I certainly do not want a BG from another nation. Multi-national should mean someone under Command of someone else. Joint-command cannot work except as a very poor compromise. I understand the real world is different, but we do have to recognise that such organisations will run less well, when under real-pressure. So-
Corps should issue order for the next 36-24 hours, which should flow down to leave BGs 4 hours to plan and issue orders. That is the gold standard required and I doubt you can do that in a multi-national HQ.

HQ Size: Well here's an issue in itself. HQ benefits nothing from size. There are endless command studies that show this.

As an aside an IDF Formation HQ is less than 100 men, all up. - It's a Signals Company. The actual key players number less than 10.