Strictly for streaming purposes, no. In fact, the base line 2.3 GHz unit would work with 4GB of RAM. More RAM makes everything work quicker, granted, but it isn't entirely necessary particularly for video streaming. You do not need the dedicated video card to run full 1080p video.
__________________ ACMT MacBook Pro (15-inch, Mid 2012) 2.3 GHz Core i7, 8GB, 256GB M4 SSD iPhone 6 - 64GB S/G Sound System Audio Engine A2 Display UltraSharp U2412M 24" Custom Built Gaming PC Kaby Lake Edition | 16GB DDR4 | GTX 1070 8G
Do you have other macs in the house? If so, it seems this is where the Apple TV would fit in well if your using iTunes for all your music and movies. I store my movies on my iMac or on my ipad 2. I airplay from the iPad and stream from the iMac, works perfect, nice, clean and easy. Any reason for a mac mini? Planning on surfing and doing other things you can only do with a dedicated computer?
If you are neutral in situations of injustice, you have chosen the side of the oppressor. If an elephant has its foot on the tail of a mouse and you say that you are neutral, the mouse will not appreciate your neutrality. Bishop Desmond Tutu