I'm wondering why is the single leg version of the deadlift is usually performed with stiff legs. For instance, exrx.net and most of the youtube videos that teach the single leg deadlift show it with stiff legs. What is the reason for this? Are there any risks when performing the single leg deadlift like the standard one?
A big portion of it is simply the mechanics of the lift. Just try to stand on one leg in a conventional deadlift stance (sumo would be impossible). You run into the big question of what to do with the leg you aren't using. If you stick it in front of you like a pistol, grabbing the bar becomes nearly impossible. If you stick it behind you there is a high likelihood that you don't have the flexibility to keep it behind you like a half split while you initiate the pull.
The second portion is understanding what the exercise is designed to target. The primary part of the posterior chain that single leg deadlifts are hitting is the hamstrings, and secondarily the glutes. As such, you need to follow these guidelines:
In the video you linked to, the leg was bent too much to be of much use on the hamstring and glutes. Instead, it will be more focused on the hips. It also loses the stretch reflex making it less bio-mechanically advantageous.
The single leg deadlift is done with a stiff-legged due to the fact that it's biomachanically advantage:
From a Sports Physical Therapist standpoint, I prescribe all single leg exercises to include the single leg deadlift for a few reasons:
As for as muscle recruitment goes, read this article. Basically, stiff-legged deadlift has been shown to recruit more hamstrings, glutes and paraspinal muscles.
There are risks for all exercises; however, with proper form and technique, the single leg deadlift is safe and effective for the above reasons. Hope this helps!