As long as you're using an amplifier the little mixer board should work by connecting all 4 resistors at the "output" of the mixer (input to the amplifier). You'll simply be mixing 4 signals together (4 -> 1) instead of mixing two pairs of signals (4 -> 2).
You should NEVER directly-short two (or more) audio OUTPUTS together, but it's OK through the 2.2K resistors. (It is OK to short INPUTS together, so if you have a mono source you can directly connect it to both inputs on a stereo amplifier.)
The downside to a passive mixer is that you have a voltage divider and the signal gets cut in half (assuming two signals, equal resistors, and equal output impedance). Mixing left & right cuts the signals in half again but only the different left-only and right-only signals. The "center channel" information that's common in left & right is not reduced. Hopefully, your amplifier has enough gain to make-up for it.
Real mixers are built around a summing amplifier. That's pretty easy to build with an op-amp but it gets trickier if you don't have positive & negative supplies. If you have a single supply, the signal has to be biased (so you don't loose the negative half of the signal) and then the bias has to be filtered-out, so that's a few more components.
Of course, receivers and other regular home-audio equipment don't mix... There's a selector switch so only one source is connected at a time.