SMR’s  Resume
First, I have to back-pedal and admit that the issues which enflame your accountants are most likely extremely valid & you are lucky that they noticed them.  My compulsion is to remove the sting & inconvenience from these issues.

So, back to the point:
You need your accountants to provide the information you need, on time, in order to keep up with your business.  Yet it seems like the accountants are always full of some urgent drama or other.

Why do the accountants make such a frenzy out of month-end close?!?
Well, for 3 reasons, really.

 They are trying to accomplish 4½ weeks’ worth of analysis during
   the first week of the month!
Use daily tie-outs instead.
The trick is to include the important information in the tie-outs.

 They need to record information that has not yet been received!
That's what estimates are for.  It takes proven discernment to correctly decide what information is needed and which amounts can be estimated, based on which substitute information.
Information sources need to make providing monthly results a priority!

 They are recreating the wheel!
Formalize the account analyses ~ this stops the need to re-invent and increases reliability & accuracy.
Standardize account analyses ~ this results in efficiency by reinforcing familiarity & consistency.