The two approaches to this vexation you refer to, fail to explain and discuss how life and will are intrinsically entwined.
The will refers to a process that funnels the various inputs one is conscious of into an output (e.g. action or thought). It is only "entwined" with life to the extent that the life in question is capable of being conscious of its inputs.
But what is important is the nature of the process, not who we hold the process to be capable of being actioned by.
On one hand we have a false analogy that appears to be adequate, claiming a thermostat as being willed and living and compatibilist that seek to accommodate such an absurdity.
There is no false analogy unless one is intent on begging the question. There is a deterministic process (the will) that converts input into action. The onus is on you to show how this is different, other than in mere complexity, with how a thermostat works.
At the moment you haven't, and don't seem able or willing to. Instead you appeal to that which we reserve the label for, rather than looking at the nature of the process.
Co determination involves a will, a living will and it is the very nature of that will to seek to self determine it's existence.
Question begging. Please try to offer something that is more than that.
Until determinists and compatabilist include the will in their discussion of free-will I fail to see how those approaches are even relevant.
They probably aren't relevant to you, it seems, because you are simply not aasking or answering any question of any relevance to anyone other than your own question-begging self, rather invoking a pedestal upon which to place life, and going: "look: life! It stands separate from the universe!"
How can you discuss Free will with out referring to the will itself?
The will, its nature, and specifically the freedom within it, is at the heart of both the compatibilist and incompatibilist position. You seem to be failing to recognise it.
Self = will
will = life
will = determination
will determination = self determination
No will = No life = no self determination
Care to put this into something more meaningful than fallacious equivocation?
Include life in your assessments and arguments and I think you would have a better argument than what you currently have.
The (in)compatibilist position considers life at the heart of its position. If you can't or don't recognise that then you're not even in the same zip code, let alone on the same page, in trying to understand those positions.
And you still aren't addressing the issues raised with your notion of "co-determinism". It's honestly as if you don't understand the criticism and are just repeating the same spiel.
Is the will a deterministic process? That‘s where the comparison lies. To dismiss the comparison because one process is not "alive" and the other is deemed to be, is a fallacy on your part, unless you can show that the nature of the process between a living one and non-living one is different with regard to being deterministic or not.
Can you do that? Are you arguing such a difference? If not then the comparison stands. Any rejection it simply because one is part of a larger process that is considered "alive" and one is not is simply fallacious.