Gentlemen!
A run-in-and-break is designed to allow a very standard arrival pattern to a runway. You run in from an 'initial' point, which is a nominal distance out from the 'in-use' threshold, and fly along the 'in-use' runway, on the runway heading and slightly offset on the deadside. At an appropriate point as you pass along the runway you 'break' into the circuit to fit in with other circuit traffic. In the military, the speed is irrelevant as it works at all speeds, the amount of G used during the break bleeds this off if necessary. Climbing, descending or staying level in the break is irrelevant. In military fast jets the reason it is done is because we fly in formations and it is the quickest and tidyest way of getting a formation to the airfield and on the ground in good weather.
The nice thing about the procedure (note I call it a procedure, cos that's what it is) is that is is very standard. When you call 'initials' anyone in the local area can look and see you in the appropriate place and they know that you are going to fly from your current location, directly along the runway, on runway heading, and out of their way, until you 'break'. The advantage it has for the joinee is that you can see all traffic in the circuit as they are all offset to one side of you as you run in to the airfield. I would say that this is not the case with the current collection of accepted airfield joins used around the world.
PS. How can you say that the overhead join is the 'standard join' when it is rarely practised if it is actually allowed at all at some airfields.