If you want to do it in the digital domain, FPGA is the obvious choice to go with here. It'd be difficult to get enough channels connected to a 'custom' design any other way, but yes the learning curve will be steep if you go that route.

There is enough bandwidth on a USB 2.0 HS port for many channels of audio, you could just buy a couple decent multichannel USB 2.0 HS audio interfaces. They are pretty readily available with 8 in / 8 out (mono) channels in one box for a few hundred bucks. You can plug these into a RPi and it probably has enough grunt to do the processing using JACK or pipewire or some similar audio router with some custom code to manage it. Full-pipeline latency could be problematic, especially if you need to stay in sync with the TV and whatnot that is pretty noticeable and often can't be compensated. This is probably the simplest way to get something that 'works-ish' though. People use similar (smaller) setups for realtime room/speaker compensation filters and it seems to work alright, and this application isn't too far off from that. In fact, you might want to do that too.

There's an analog solution too, your RPi or microcontroller can manage an array of variable gain amplifiers (using a slow / cheap multichannel DAC) or programmable gain amplifiers (directly with I2C or SPI) which form the inputs to a traditional summing mixer circuit. Then you don't need to worry about processing some dozens of megabits of audio data with low-ms of latency and just need to set appropriate gains, but the analog design is a bit more complicated. Still, this is IMO much easier than an ADC -> FPGA -> DAC pipeline.

Prototyping SMT parts you can either buy protoboards with the chip mounted and pins for use with a breadboard, SparkFun and AdaFruit make tons of modules like this, or try AliExpress, and you can find devboards for pretty much any microcontroller or FPGA too. If the chip you need to work with doesn't have such a board available, you can get generic breakout boards for most SMT packages and solder the chips on yourself. It's not that difficult, though it takes decent equipment and a bit of practice. Having custom PCBs made is also super cheap now, and that is how I would recommend approaching this, if you're willing to tackle the soldering challenge and don't' mind the somewhat long iteration cycle as you wait for boards to arrive from Asia (or don't mind paying for premium shipping or a more-local manufacturer).