Hello Vorg, very welcome to the Delta Throttle community.
There are several tricks and solutions available to get multiple buttons working on an arduino-based joystick, ranging from the use of shift registers and I2C i/o expanders, to simply upgrading the electronics to something with more native i/o pins.
I'm going to wire my delta throttle (once I get the basic mechanics figured out) to an Arduino Mega equivalent board, giving me access to 54 digital and 16 analog i/o pins.
At that point, I'm already past what joy.cpl can handle for a single device, though there are ways to get more axes/buttons to work.
I'm looking into putting together an alternative handle, possibly modeled to match the Warthog joystick. That should provide enough controls at your fingertips.
I'm thinking to put an analog ministick in place of the CMS switch or possibly the MMC.