It seems like we are having the same issue on some machines - particularly the busy ones in library areas.
The Macs are: 21.5 inch iMac 14,1 running OS X 10.11.6
I haven't yet looked at this issue myself - my collegues have.
Though I can't think of anything that I would have tried - that they have not already tried...
Personally I have a suspicion that it's related to logouts not happening correctly - getting stuck,
and then then machines being crashed, in order to log in..
Quite why logout sometimes does not complete (apart from applications still being open, with unsaved documents) is unclear.
I did think of writing my own logout routine to 'force' a full logout - even if that did result in loosing unsaved documents..
but haven't yet done this since I've been busy with other things.
besides implementing such a thing should not be necessary..
But it would be interesting to see if this then began to resolve the startup issue..
If the problem was related to 'corrupted boot caches' - then the "safe boot" followed by restart and "normal boot" may sometimes resolve this - and I think on occasion has for some people. Although that method is not always 100% reliable.
What we know is that 'something' or several different 'somethings' are on some machines, causing the system startup to not follow a normal pathway - resulting in a startup freeze.. And at this point no-one seems to know exactly what is causing this.
Reformatting and Re-imaging the machines affected - does resolve the problem - for a while, and then it occurs again.
Although above I said re-imaging - we are thin imaging using Casper, and installing an OS from deploy Studio (which we didn't do last year)
but last year we saw this problem too - though less frequently. The OS is built using the AutoDMG tool..
So it's not an old-style 'clone' image.
The main 'pattern' so far - is that we are only seeing this happening in especially busy areas - where lots of different logins are occurring..
i.e. - in excess of 100 different users logging in..
Incidentally, in casper Vn 9.96 the number of: "MDM Capable Users" (Machine Record : General ;MDM Capable Users)
does not seem to get reset after reimaging - it just increases with occasional ,, entries which makes me think that might be a casper bug ?
But even if so, that unlikely to be related to this startup issue..