The Musicophile

Ideally you'd use an I/O unit with the required number of tracks (I've had some really good results with a MOTU 24 I/O, although this is might be overkill for you) and a nice mixing desk, with as many mic inputs as you require.

You'd connect the tape outs for each desk channel to the Input side of your I/O unit, and the Output side of your I/O to the desk's tape returns. The I/O unit converts the analogue signal to a digital one and sends it to your computer. Your computer records the signal, in whatever DAW you choose. When you play back the audio the computer sends digital signals for each channel back to the I/O unit, which converts the digital signal to an analogue signal, and sends it back to the desk. This way you can do all your mixing on the desk .

The desk gives you a lot more functionality than your typical audio interface (effects sends on all channels, fully featured monitoring, you can do your mixing on the desk, etc), provided it's a decent desk.

The downside is that you'd probably have to spend more money than you would if you just tried to find an audio interface with enough channels.