Well as far as this question goes, anything can be answered. From one word to a whole thesis.
The more complex a system is, and the higher the probability of failure. In general. So we could say that a microcontroller + firmware is per se a much more complex *system* than a simple, purely hardware-based feedback. Now of course with a microcontroller, you can implement features that would otherwise be clunky to implement, and that can significantly reduce the number of parts in your design. System-wise, the microcontroller with its firmware is still more complex, but it can reduce the number of parts and thus fewer potential points of failure in terms of hardware.
The choice is, you guessed it... engineering.
To address the specific problem with your client, that's a very common issue IME. I dunno about you guys in general, but clients who think they know better than you on some technical points (while needing your expertise to design), often based on preconceptions ("I think that...", "Several people in the field have said that..."), they are not rare, and they are usually a pain to deal with. But in the end, it all comes down to your communication skills: convince them while still conveying the idea that it's in their best interest.