Which desktop is better for Console Streaming?

I have an ACER All-in-one touch screen that is pretty bad and unfortunately my parents purchased another all-in-one this time from HP.. The acer I can not upgrade so hopefully I can upgrade this HP.. here are the specs for the HP:

Keep in mind I only want this PC to be able to stream my CONSOLE gaming.. I do not play PC games.

I copied and pasted this from their website sorry if it is hard to read, thanks in advance.

1.6GHz Intel Core i7-720QM
Windows® 8 Professional 64-bit, Windows® 8 64-bit, and FreeDOS
18.5-inch diagonal widescreen HD LED-backlight anti-glare LCD Display
Integrated All-in-One form factor
Adjustable tilt stand
AMD A68 FCH chipset supporting E1-1500 processor featuring AMD Radeon HD 7310 Integrated Graphics
Integrated Realtek 8175EH-VB-CG 10/100 Ethernet Controller
Wireless connectivity - HP WLAN/Bluetooth Combo NIC 802.11b/g/n Mini Card
Optional Integrated 1.0 MP 720p low-light webcam with dual microphone array
Integrated HD audio card and stereo speakers
Up to 8GB of DDR3 SDRAM supported
Up to 1 TB hard drive
Optional HP Slim-Tray SuperMulti DVD+RW SATA 8X Optical Drive
Media Card Reader
USB Keyboard and Mouse
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  1. Oh and my current PC specs:
    Model: Acer Aspire Z3731

    OS: Windows 7

    1920x1080 resolution

    CPU brand - Intel
    CPU family - Wolfdale
    Base CPU model - Pentium E6700
    Cores - 2
    Clockspeed - 3.2 GHz

    GPU (base)
    Graphics type - Integrated
    Integrated GPU brand - Intel
    Integrated GPU model - GMA X4500 HD

    Base RAM size - 4 GB
    Max. config RAM - 8 GB
    RAM type - DDR3
    Base RAM speed - 1333 MHz
    Slots - 4

    Base storage type - HDD (5400RPM)
    Base storage capacity - 1 TB
    Card reader - Yes
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