I can't comment on the Pi itself and why it's unable to keep up. If it is an I/O bottleneck then I'd have to wonder if it's the ports or whether a scary-fast SD card would solve your problem.
In terms of a new system, though... IMO, I'd avoid MSI if you're planning on running Linux - there's a kernel bug that doesn't play nicely with USB3, and MSI doesn't allow you to step the USB3 ports back to USB2. While it may work for you, there's no effective workaround if it doesn't.
Regarding the setup, you'll find that the motherboard is one of the biggest power drains on a modern system; current Intel processors barely use anything at idle, but you'll have a background consumption of 20-50W just from the mainboard. So think about what you really need there, and whether you can switch bits off to save a watt here or there - as a rule, fewer chips means less power, so form factor is a good indication of power requirements. In whatever sized board, something with 2 x RAM slots, 1 x PCIe and 3 x USB will be lower power than something with more ports than you know what to do with. Of course, you still need enough ports/slots to do what you want to - keyboard, mouse, UPS, tuner...
A Celeron system would have more than enough grunt to record and stream. Indeed, I used to use an Atom, as simply moving the stream from tuner to LAN, tuner to disc or disc to LAN really doesn't need much CPU. I would assume that this will hold true as we move to H.265/HEVC, although that's hardly mainstream just yet but it may be something you'd want to consider. However, if you want to process the files in any way - transcoding, specifically - then you'll need something much more powerful. If you just want to filter out an audio stream then that's much easier on the CPU as it's basically the same effort as re-wrapping into a different container.
