There are lots of DIY projects of flight simulators who have built whole cockpits around the PC hardware. That goes from simple I/O such as switches, potentiometers and LEDS, to reused old aviation equipment.
The amount of I/O is never a significant problem. There are simple ways to expand, such as with 74hc595 shift registers, or you can dedicate an 8 or 16 bit port as a databus, another port for an address bus and some more I/O pins for read, write and chip select signals. This way it's easy to get to several hundred or so I/O pins. More too, but latency or bus buffering becomes more of an issue. And as BruceHolt already mentioned, microcontrollers with a lot of I/O pins are readily available, even up to 160 I/O pins or there abouts. Working with a data / address bus does make the system more flexible (when using connectors). You can for example add modules later.
An FPGA can have more I/O pins, but it's probably overkill for this, and too complicated to set up.
Another approach is to use an industrial bus bases system. With for example modbus, you just need to add an RS485 driver (costs around EUR 2) and a bit of software to your uC, and then you can put a whole lot of uC's on that bus, and control them all from a single location. Main bottleneck here is throughput. With older 8-bitters, 115k2 is around the top for a "sensible" baudrate (the uC's also need time to do something with that data). With more modern uC's, 3MBaud is around the top what still makes sense. Such a bus does have quite a lot of overhead. Nearly half the data sent is for addressing and checksums, and you normally need two packets (command and response) on the bus to do anything. CAN bus is also an option.
If you want more throughput, microcontrollers that support ethernet are also available, and there are boards such as the beagle bone black. It runs linux, (including ethernet stack and more) and has 70+ I/O pins natively. The Olimex A20-OLinuXino-LIME2 has 160 GPIO pins. This does necessitate using connectors with a smaller pitch and that does make it more fiddly to work with. This board also has Gbit Ethernet, but due to some hardware bugs, the throughput on for Ethernet is a bit disappointing. And there are many other SBC's too. Main thing here is that you to realize that using an SBC as "just an I/O expander" is a viable and affordable option. These linux boards can easily run without a monitor or keyboard attached and then be dedicated to a single (or multiple) user applications.