Hello Dave,
I assume you mean that the computer with a 12th-generation i9 CPU is a 'mini' form-factor PC, without PCIe slots.
Graphics cards don't usually have audio outputs, and Hauptwerk isn't demanding in terms of graphics capability (any modern computer's built-in graphics chips should be fine, unless you also want to connect several high resolution, e.g. 4K, monitors simultaneously).
Although most PCs do have a built-in audio output, they aren't ideal for use with real-time/low-latency music applications like Hauptwerk. Sometimes the freeware ASIO4All driver can achieve acceptable low-latency performance from them, but a dedicated professional/semi-professional audio interface, such as the Focusrite one that you mentioned, would always be highly recommended (potentially much better low-latency performance, better audio quality, plus built-in MIDI capability, plus the ability to use multiple audio outputs for multi-channel audio if desired).
You can attach USB audio interfaces (such as that Focusrite) to the USB ports on the PC -- the PC doesn't need PCIe slots for use with USB audio interfaces. It appears that the Focusrite model you mention has a USB-C, so you would need your PC to have at least one free USB-C port to connect it to.
Hauptwerk certainly benefits from plenty of CPU power, although the theatre organ sample sets (such as those from Paramount or MDA) are dry, so not especially demanding. Some large wet/surround classical organ sample sets do need powerful CPUs, for which a 12th-generation i9 CPU should be ample. If you're only using theatre organ sample sets then you probably don't need quite such a powerful CPU, but it certainly wouldn't hurt.
I hope that helps.