If your camera required just the pressure of a button to take a photo it almost certainly was capable of metering, setting aperture and shutter speed all by itself, just like digital cameras; you just didn't have access to those settings, because of the simple design, intended to lower costs while making it user friendly. All of this also implies it had some electronics inside.
Compact p&s film cameras usually have very simple center-weighted metering and a small range of aperture and shutter settings, however this is enough to handle most scenes in daylight. It gets much worse at dusk or indoors, I'm sure you noticed :)
Can you remember if your camera needed some kind of battery? If film advance was hand operated you probably wouldn't need to change battery for a very long time, and a simple camera like that could really look like it had no electronics.
edit: I'd say it's probably most common for very simple film cameras (at least in the last decades) to have a single aperture and a narrow range of shutter speeds, with some electronics for metering and setting the shutter. I remember some almost-toy cameras with switches to select sunny or cloudy, that might have been a way to control exposure, probably altering shutter speed.