It's not about vague things like "opponent quality", it's about what they actually did while they were in the cage. And not only does the single-leg have a proven history of not working as well in the cage cuz' the cage interferes with the mechanics of the move (singles are all about getting your opponent off-balance through either turning them or getting a different angle they didn't expect and can't sprawl on, and it's way harder to do that when they have something to lean against, like a cage), but-- and this would even happen in the fights Kawajiri would win too, by the way, and against guys like Jason Knight who don't have the wrasslin' of an Olympian like Kazuyuki Miyata, either. Every time Kawajiri would try to use his favorite single-leg moves, he'd have to work way harder to complete it (if he'd even complete it) because, for example, he'd shoot for an outside-single, transition to a high-crotch, wrap one of his legs around the leg he shot in on, then transition his arms to a waist-lock and, as he'd use his ridic' strength to rotate their bodies and (with his wrapped-leg locking it in place) throw them off-balance to secure the takedown, ideally landing in half-guard, their bodies wouldn't rotate because they'd start pressing their shoulders into the cage to keep themselves upright. He did this in just about every fight he had in the UFC, and that happened every time.
The Dennis Siver fight was the best example of it. Kawajiri'd get a single and drop down to work on it, like he always does, and Siver would just press his body against the cage to defend it and they'd be in a stalemate, but Kawajiri would be the one wasting more energy cuz' he'd be working on the single-leg while Siver was just keeping himself upright. It wasn't until Kawajiri started to switch it up and transitioned to double-legs against the cage that he was able to take Siver down, and even then, because that's not really Kawajiri's style, they looked awkward and unclean compared to a lot of other guys who use 'em.
If you live near a fence, try this out: lean against the fence and force a lot of your weight into your right shoulder while pushing it to the left, without letting your left shoulder lift from the fence. Pushing almost enough to to force your left shoulder off the fence and rotate your body. Then lift your left leg off the ground. Then imagine how you'd be falling on the ground if the fence weren't there. It gives you an idea of the situation.
A lot of shootfighters who fight in a cage who have been able to show a good single-leg have all had to develop really idiosyncratic methods of utilizing it so that it doesn't get nullified. Like Shinya Aoki; he uses his height and ridiculous grip strength to both pull up the leg as he forces his shoulder into their hips to drive them down, landing himself in a leg-ride, then he uses his legendary top-game to transition to half-guard or mount, then to their back. And Daniel Cormier, who uses the strength that comes from his wide center of gravity to get lifts off of a single-leg and slam his often-much-larger-opponents, like in a lot of his most famous gifs, and works as a chain-wrestler outside of that. And Ben Askren-- his nickname "funky" isn't just cuz' of his hair, it's cuz' he's a master of funk wrestling (look it up.) I'd include Masakatsu Ueda in there, but I haven't watched enough of his cage fights intently to really feel like I can include him (the fights of his I have watched intently were determined more by his striking-- roundhouse kicks to the body and 2-3s eff-tee-double-you.)
Otherwise they tend to just abandon 'em and keep it as an ancillary tool, relying primarily on double-legs, like Chael Sonnen. Or Georges St-Pierre, perhaps most notably (and he didn't even come from a wrestling background.)
The cage is also why double-legs aren't as effective in a ring or in real wrasslin' as they are in caged-MMA, which's why you see 'em so much more in the UFC or Bellator than you ever did in Pride or DREAM or Shooto or in any wrasslin' tournament. The ultimate defense to the double-leg takedown is a sprawl; it's, literally, the way you shut down a double-leg takedown. You time it out, sprawl your legs out while pushing down on top of your opponent's body, like a cat when you drop it on the ground, and they're forced to either abandon it and re-shoot or to use the close space to transition to a single-leg and get some other stuff going (Ben Askren does this a lot.) And, you know, you can't really sprawl your legs out when your back's to a cage. You drop down and shoot your legs out... and they just hit a cage. They didn't shoot out. So you're in that awkward position when your legs are half-bent and you're kneel'd down and look silly, then the opponent jumps on top of you. Which, for obvious reasons, doesn't happen on a wrestling mat or in a square ring.
A lot of guys that are known for having good wrestling don't even try to shoot if their opponent's not cut off into the cage. If their opponent's not charging forward, making it easy to just time it out and drop down, ala Georges St-Pierre or Demetrious Johnson, they know how likely it is they'll just sprawl and win some points while they're on top of you and you feel silly for wasting that much energy and having some guy leaning on top of your back, so they don't even try to do it.
Birchak was also a 4-time Greco-Roman state champion in high school, so he was a legit wrassler, and, considering the entire fight took place in that domain and was about Kawajiri's wrestling vs. Birchak's, that's a very relevant point. Birchak also holds a knockout victory over Joe Soto-- who was a Bellator champion and a top-15 featherweight during that time, and had just come off a very competitive title match with TJ Dillashaw and is currently on a three-fight winning streak in the UFC-- and he'd left the UFC off of a win, so he was more than just a middling guy. Considering how much noise-making in the UFC is about shit-talking and being in brain-damage-causing-wars-- which apparently doesn't even factor in that much once you get beneath featherweight, and still doesn't matter that much if the guy does one or both of those things but came from Bellator-- using the amount of noise made in the UFC as a basis for quality is a real silly way to determine a fighter's intrinsic value.
