You've stated that the 'Inspector Generals Board' as you decided to call it would be charged with;
"creating new training and combat doctrines, as well as helping with the development of their respective arms, such as by assisting part of the procurement committees in charge of getting new equipment."
In reality as someone else has noted those two tasks would fall within the remit of the General Staff itself (ultimately every facet of military planning everything does). However development and procurement are two very distinct and specialized areas of expertise and are therefore usually separated into two distinct command structures within the military. So while both branches would ultimately accountable to the General staff they would not be combined in an Inspector Generals Office.
In the real world however the title of 'Inspector Generals' (IG's) exists and serves a very specific and important role/function (Hint the purpose is in the name.) And that is to ensure the projects being managed by the other two arms are actually progressing as reported back to the General Staff.
It's one thing to be told by your weapons development branch that 'project x is performing to spec and will be ready on time' or that 'the army's order for Y is being delivered in quantity, on schedule and within budget.'
It's entirely another for everything to be going exactly according to plan or if its not, finding out exactly why not. That's where the IG's Office comes in. They are teams of subject matter exerts in fields like engineering, electronics or logistics and accounting etc that get sent in to audit and inspect programs to make sure everything is running smoothly. And in cases where problems have been identified and reported upwards to the General Staff they can also;
- review and analyze what the issues/problems are,
- determine whether the best solution has been chosen to fix the problem;
- come up with suggested fixes; and
- if nothing else identify who or what is to that went wrong and whose blame!
So they can both give the General Staff independent oversight of important projects and equally important if things are going wrong they also have the ability to sit down with the other two branches, ask hard questions and come up with possible solutions that can be fed back to General Staff for consideration/approval.
They won't always be popular with the people they 'inspect' but they will be (and in the real world are) essential to the General Staff.