One of the big points is that looped architectures don't necessarily remove CoT style reasoning. They definitely change it, but the model can still be prompted for this kind of output. It just spends more computation on individual tokens. I think the hope, in that particular scenario, is that the model spends less tokens on things like backtracking.
But there are lots of approaches to looped transformer architectures. Some have some kind dynamic loop (i.e. they might loop until they satisfy some kind of equilibrium, or they might have some kind of routing function that makes different tokens loop differently, or any number of other approaches). Others are simpler "loop this many times" architectures. It's a really active area of research with some pretty deep roots, if you want to dig in! This paper blew up a while ago, and it's a looped transformer architecture (though not like the ones discussed here) that isn't autoregressive, as an example: https://arxiv.org/html/2510.04871v1