i was wondering wether a HEAVY overhead press variation (military or push press) or the bench press will develop more functional pushing strength for strongman events (obviously if you can bench 500lbs youll be able to log press a fair amount of weight)
I'm not sure I understand your question, but it seems like you're asking if improving your barbell push exercises, eg, military, push press and bench will carry over to the log? Do I have it right? If so, it depends where you are. All of these will have carryover to the log at some point, but you'll hit the point of diminishing returns - ie, getting your bench from 200 to 300 will have more of a positive effect than getting it from 300 to 400.
There's no real question to answer here, and certainly not enough info to answer it specific to YOU, but in general, the best way to increase your log is to do the log. If you dont have access to one, you make do with what you've got.
There really are too many variables to give a direct answer to your question.
In a lot of cases improving your timing, technique, or leg power will provide better carryover than simply increasing static strength in upper body lifts - especially if you are talking about bench press.
For the log I would say that incline dumbbell press or medium grip incline bench press will give you the most carryover. For an axle I would suggest focusing on standing barbell push press. If you are going to bench press then I would focus on close grip bench press. Dips (weighted) are also good for building pressing power.
If there is something specific that you would like to focus on I can give you a better answer in terms of setting up a game plan to improve that particular movement/etc.
I hate to bust your bubble my man but being able to press 500lbs in the bench press does nit even almost guarentee even a decent log press. I have been to many comps where some huge bodybuiloders and even powerlifters for that matter have been put to shame by even 270 lbs log and even lighter for that matter.
My advice which I am a nobody in the strongman world but I have observed and learned and I am half way knowldegable. The front squat and standing strict press will honestly help you more for the log and axle more than any other exercises you can preform other than the log its self oviously.