There have been several two and three button joysticks for the C64 and games have used these feature.
It could have been done
As Retrograd (and others) already explained, the basic design of the control ports is made for either
- a digital 4 direction plus fire joystick or
- two (analogue) paddles with one fire each or
- a light pen with one fire button (only on port 1)
So in theory an analogue joystick with up to 5 digital inputs could have been used. Much like the Apple II did (or later IBM).
It wasn't and the Reason is History.
It's important to keep in mind that the joystick port design is not a Commodore development, but taken straight from the Atari 2600 and used for the VIC20. Doing so gave Commodore the advantage to skip developing their own peripherals and straight buy ready developed ones, just to be branded. A classic Tramiel move :) The C64 again inherited the port from the VIC20.
Atari did not offer any analogue joysticks for the VCS, as polling them would have been rather cumbersome for the limited VCS hard/software without much gain. Similar not third party developer did.
Hen and Egg
Later machine, like the 400/800 and VIC20/C64 could have done better, but it's a classic hen and egg problem:
- No need to write games for analogue joysticks without them
- No need to offer analogue joysticks without games using them.
Essentially this would be the end of the story, except, like you already assumed, there was a strong need for more buttons so any solution would be welcome.
Atari to the Rescue
With the 7600 Atari introduced the CX60 controller, a new two and a half button controller with a left and right button, plus a third, virtual button when both are pressed (called Command). To do so the analogue X/Y inputs were used as digital inputs for both new buttons, while the original button input was connected via two resistors. Worked fine with the 7800 as well with the 2600, where it later was called CX24.
While basically compatible with the C64, these joysticks would not work straight away as the 'main' button needs pressing both. Still Atari paved the way to follow by opening great possibilities for third party development.
Not everyone liked the small joystick. And having a trigger at the stick sounds as well great. Multi system capabilities are a great plus, as the same product can be sold to more customers, which can easy outwight the additional manufacturing cost. So configurable joysticks it was. Working with the regular main button system as well as with the modified 7800 and offering up to three buttons for major systems.
Like any add on manufacturer joystick makers were keen on new features and more buttons (like auto trigger before) would add a great sticker on each box: New, Better, More!
With these joysticks becoming more common, game developers also started to follow with games using them.
The Rest is History
In the end, even Commodore used this method for the C64GS Joystick, which added a second button using the paddle A (POTX) input. Later the Amiga sanctioned the full 3 button joystick with using paddle B as well.
At that time the three button layout was well established. For example with the Kempston Interface for the ZX Spectrum. Atari used the 2 button configuration of the C64GS.
But There's More And You Knew About It
or by your foot!
There was a controller with 5 large buttons to be operated by foot - so it was quite possible to add more control via the second port, at least for single player or hot seat games. :)