When ever someone says any chase emergency would just be running or just be speed vaults I think they lack experience and imagination. I grew up out in rural community with allot of independence and had some scary running away situations happen not mention playing tag and chase in the woods allot. Being able to do complex movements that require agility and co-ordination is important. It is also my opinion that generally being able to run far is not nearly as important as being fast, Nobody starts out a race at 8 minute mile pace its always a sprint first if you can quickly put distance between you and your opponent they will usually give up.
When I was 12 my neighbor caught me stealing adult magazine from his shop, I bolted, ended up running about hundred yards through the woods which included jumping over a fallen tree scrambling through bushes and jumping over a bike jump before he gave up the chase. Of course he was my next door neighbor so my victory was short lived as he just waited for me at my house and chewed me out there when I slunk back.
Another time I found a cabin in the woods with some friends, which turned out to be scarily filled with rifles and handguns, after poking around these guns for awhile we had gotten pretty freaked out, we heard a loud noise and bolted, the cabin was on a slope and had a back porch facing the slope we were afraid the guy who owned the place was coming through the front door so ended up taking the drop down from the porch 6-7 feet then sliding down this slope scrambling through bushes down to the stream we had been climbing up running down that bouncing rock to rock till we got the too logging road that had brought us there.
In second grade I got the brillant Idea to assault a 5th grader with 4 buddies for fun. They jumped on his ankles while I jumped of the big toy and tackled him and punched him around the ears. A teacher caught us and we all got detention, the next day though he caught me alone on the playground and started chasing me, I ran to a set of concrete pipes that were too narrow for bigger kids to slid into it and waited the him out untill recess was over, after that we ended up talking in detention and becoming friends.
Finally as bouncer I had to chase people down a few times the obstacles there were people usually then it was all about dogding and maneuvering and screaming for them to move but once a patron got pissed of and kicked out a window next door and I really thought I was going to have to vault something to catch him as I was anticipating him turning behind some concrete blocks, sadly he was to slow and I caught him before he could put the object between us,

I really wanted to kong to tackle that guy.
There are arguments that flips can benefit air awareness and body control and may be usefull in some pretty rare situations. I think those arguments are not well supported but learning a few flips certainly won't hurt your PK.
I have done time trials through numerous enviroments played games of chase and tag and never once while trying to adapt and move as fast as possible through any enviroment has a flip seemed like a good idea despite having a background in gymnastics prior to parkour. I did have one athlete I trained who would throw double legs in courses with objects that were to soft to vault over and to high to jump(the hedge/barbwire idea) however he was very good tricker with minimal parkour experience I suspect had he spent the same attention to training basic jumps as flips he would have been able to jump over those objects with better control and safety then a flip offered. This the problem with ryan doyle example to if you train flips way more then jumps there might be situations that they're more usefull if you don't bias your training then I think that is pretty unlikely.
In any event I prefer to focus on a positive definition of parkour not a negative one, train to be usefull to be able to overcome obstacles as effectively as possible. If you think about that honestly the majority of your training will be simple movements, running, jumping, climbing and moving on all fours. If you do a flip now and then great if you find yourself focusing mostly on flips I don't think your doing Parkour anymore then a tricker is training for a streetfight whether the tricks are martial arts based or not.