It's much easier, faster, maintainable and usually cheaper to add a cheap controller and peripherals than to find an electrical engineer to do the math and design a circuit.
Analogue systems have a lot of non-ideal characteristics such as drift, thermal sensitivity, parasitic capacitance and inductance, impedance matching problems, insane cumulative errors and manual tuning requirements that don't exist in the digital domain. They are also more expensive to modify, and the price goes up significantly with complexity and speed. Not to mention small and fast is really difficult in the analogue domain thanks to everything being a transmission line.
Any sufficiently advanced analog computer is indistinguishable from a noise generator.
Modern autopilots are all sensors, DACs, ADCs, and various digital computing resources. Cheaper, faster, more featured, and mystifying in their failure modes.
Although I am a big digital person, even for photography, audio and other seemingly analog things - I admire and appreciate the one major analog computer in my life.
Alas, after 7 years, it finally needs a repair. Sigh.
The altitude is “held” using a “static” pressure sensor and pitch is adjusted in a control and feedback loop to attempt to keep that signal constant. When it works, it is very smooth.
The system is not as a whole redundant, but there are many failsafes. You can put it into a mode where the system does not fly the plane but tells you what it thinks you should do. This is called a flight director. You fly it almost like a video game, get the pointers aligned.
Each component has some redundancy. It has a self test, an annunciator panel saying what mode it is in, a mode control panel, several controls on the yoke including a kill switch, several circuit breakers (one, the electric trim, which has a big red flange on it - see the recent 737 crash artickes), etc.
The workflow of using the autopilot is always command a change, check that the change is reflected on the annunciators and that the plane is adjusting attitude and such appropriately, then resume the normal scan. If at any time anything looks off my instant reaction is always to hit the big red button and take the plane’s control back. When you do this you need to be wary for an out of trim condition and other sudden forces on the yoke and rudder.