Oops… I’m not sure if this would be the right one (spontaneous: no, it isn’t).
Let me tell you my story:
I began webcasting with one of the first Yamaha 12 mixers, now called MG12 (which ist still a good mixer to start with 
)
Then I switched to my favourited Allen&Heath, a ZED-14 (which is now sold as a ZED-12 in the store you linked).
Many private user prefer the Behringer DX2000, which fits into your budget, but from my personal point of view … okay, it’s your decision. Personally, I would not spend money on that device and save it to buy another mixer later. But you will find enough user being happy with it.
Please don’t forget that you need a multi-channel sound card (5.1 = 3 stereo channels or 7.1 = 4 stereo channels) and they are either very cheap and have no good quality or expensive or they are - on the second hand market - rare.
In the german section of this forum, I started a list of devices I have found in the forum used by mAirList users:
An all-in-one device, the D&R Webstation (small mixer, multi-channel soundcard integrated, remote control with mAirList (needs an upgrade)) starts around £851 - just to show you that the price scale is unlimited… 