No, not like a thermostat. A bacterium is a compact and exceedingly complex living system. It has autopietic autonomy like we do. The fact the I navigate each morning to the refrigerator to get milk for breakfast is not so different from a bacterium following a gradient toward its food. Bacteria have complex receptors like we do that are their sensorium—-a nano-brain in the words of one scientist who studies their behavior.

And yet, also a thermostat is doing nothing else but following a gradient towards a desired state (albeit the gradient is discrete and it's moving along the dimension of temperature).

Except that a thermostat doesn't "follow" anything. It is basically a binary response state: on or off (or high and low). Nothing further. Analog or digital, it doesn't follow anything.... It reacts instantly at a setpoint. Would you think an alarm clock is "following" the time, and turns the alarm on at the set time? Or is it a momentary switch at a setpoint....