Compatibilism does not maintain that free will is the only mechanism of “choosing a specific option”.
free will, whatever it means, doesn't seem to directly have connotations of
complexity or intelligence beyond a basic decision taking capability
A radioactive nucleus may decay every moment. It has two choices - to decay or not to decay, evaluates them with a goal of compliance with the laws of nature and chooses a specific option. But it is me who sees the options, the goal and the choice in the behavior of the nucleus. This is just one way of describing what the nucleus does. But I won’t call the mechanism by which the atom “decides” whether to decay or not - “free will”. Something is missing here.
An apple falling to the ground doesn’t appear to have any choice. But suppose it falls on a pointy obstacle, now it has some options! It may choose to go left or right of the point, or it may decide to get punctured. It’s actual “choice” though is entirely determined by myriad of fine details about itself and the point. Again, even though I use the metaphors of choice, they are just that - a way to talk about an apple that ignore the details about it’s situation that I’m either unaware of of deliberately ignoring.
A chess automaton goes through the motions as you describe and lends itself to the options and decisions jargon. Just like a randomly decaying nucleus or chaotically falling to one side of an obstacle apple, it doesn’t “really” choose what it will do. But we may choose to describe it as making decisions, because such high level description is easy to understand for someone who’s used to thinking in terms of decisions.
Finally we have our own case, as choice making entities sharing the Universe with nuclea, apples and chess software. Unlike our random, chaotic and deterministic co-inhabitants of the world, we consider the free will language as applied to us literally, not just metaphorically. We are where and for whom this language originated.
There are some processes within us, that aren’t covered by free will: there are some decaying radioactive nucleas in us - which we consider our atoms be virtue of their constituing of our bodies. There are some chaotic processes that involve to many variables to be able to consider: what exact effect would a particular chemical cause in some organ when ingested in this particular body? But when we face the consequences of such injestion - say tummy ache - we describe the ache as ours by virtue of our body (and not someone elses) being aware of it.
A compatibilist free will is the feeling that we have of owning "decisions" made as a result of deterministic causes and effects. As far as we're aware nobody's programming any such thing into chess programs - because there doesn't appear to be any reason to. Would a decision-owning program play better chess? It isn't obvious. Moreover, it is unclear how would one go implementing the sensation of ownership in software.