Well, horse, there are ways of incorporating a fire here, as well.
As you can see, this shelter does a better job of managing heat loss through convection.
A fire bed used in combination with this shelter works well. Then again, burrowing into debris in combination with a fire-bed works well.
I'm not saying a lean-to doesn't work well, but it is not efficient at heat loss mitigation, again, through convection.
In a cold environment, I would use a shelter of this type long before that of a lean-to, provided I had the material at hand to make it.
Perhaps it is the different areas we live in, or even our respective level of experience, but I can tell you a lean-to doesn't hold a grain of salt to a debris shelter or even an a-frame shelter.
Also, fire management throughout the night, with the lean-to set up you describe, can be taxing and calls for a sleepless night. YMMV, however!
Alan