BJT Amplifier vs Manufactured RF Mixers
Yes, it will indeed mix anything. Unlike a proper mixer the output won't be the product of the two signals you want, but also anything else that came into the RF port. I exaggerate, but only slightly. 

You'll get mixing but not just from your RF and LO frequencies. For example if RF = 900 MHz and LO = 1000 MHz you'll get the wanted IF output on 100 MHz.

Trouble is, RF signals on 800MHz will mix with your wanted 900 MHz signal; as will unwanted signals at 801 and 901 MHz, 802 and 902 MHz etc etc, and the sum of all these may very well drown out the signal you are looking for.

Proper mixers will do this too if you operate them at the compression point ... which is basically why you don't.

Now if you don't care about interfering signals, or know there aren't any (or they are much weaker than your wanted signal) go ahead with a simple non-linear mixer. But most people do care, which starts to explain the price of decent mixers.