I've not heard of a low temperature broody but I have one theory for you. Since her first clutches had issues and one weakly chick hatched, I think we can assume she's not too low temp for successful brooding --- but maybe she's got an issue like something that went through my flock one winter.
An aggressive bacteria was brought in which killed any eggs it infiltrated. It could do this via the touch of seemingly clean feathers on a hen that had touched an infected egg, or via infected eggs touching other eggs, and was virulent enough to cause severely fast rotting of developing eggs. It killed chicks from healthy eggs as they hatched, so it worked extremely rapidly. Infected nest materials also spread it.
(I did a lot of 'hot potato' egg and chick swapping as I determined which first time mothers weren't up to scratch and so needed to share their babies with proven mothers also brooding at the same time, or for whatever reasons at the time)..
If your hen had eggs bust on her feathers from previous clutches and went right back to brooding within the next few weeks it is possible she is carrying a pathogen that causes some sort of toxicity syndrome in eggs. With me, it was a one-time random issue with no noticeable precursor nor follow up results. I noticed which hens were killing eggs by their very touch, and which eggs were dead or dying, and stopped the cycle via stopping the hens brooding and throwing the duds.
An egg that's a dud, no matter the stage of development or reason why it's a dud, stops metabolizing and the gas exchange stops working like it did when the egg was alive. This very quickly leads to a cooler surface to the egg. A smarter hen will not sit on nor attempt to brood any egg that does not quickly warm to her brooding temperature, because it's invariably dead. I've had chicks hatch from eggs abandoned every night in the dead of winter, let to go stone cold --- but you can tell which are alive because when you hold it, very, very quickly it begins to heat up as it kicks into action. Duds refuse heat and reject it, and won't hold temperature even when a hen is sitting on them.
For my less instinctive hens, I can weed out duds reliably now just by holding them a minute or so. For more instinctive ones, there's obviously no need. Off topic, kind of, this is why some hens will not sit on golf balls or hollow plastic eggs ---- their instincts say it's about to explode because it's far gone.
Hope this was helpful in some way. Best wishes.
