Customizer Not Applying Any Change, No Error Thrown

So I finally have it. The problem came from the php.ini configuration option filter.default. The option was set to:

filter.default = full_special_chars
filter.default_flags = 0

As explained in the PHP documentation, this had the effect of sanitizing and stripping characters from all $_POST, $_GET, $_COOKIE, $_REQUEST and $_SERVER data, effectively beating me in the back in a very vicious way. I have commented out these lines to set them back to their default:

;filter.default = unsafe_raw
;filter.default_flags =

I hope this will help someone in the future as I do not wish this type of unproductive treasure hunt even to my worst enemy.