You can not use motherboard HDMI with the dedicated graphics card.

What GTX 1070 do you have? MSI, ASUS, Gigabyte etc? By reference design it has 1 DVI, 3 DisplayPorts and 1 HDMI port.
You should have more than 2 HDMI outputs from the graphics card. DisplayPort can be converted to HDMI with inexpensive adapter, DVI as well.

If you have vMix 4K or PRO, you get 2 Fullscreen and External outputs. So you could use 2 outputs (PGM, GFX) from the graphics card, but this way you no longer have multiview (maybe not needed now, but in the future..). And use for example DVI for system monitor.

For the program output I'd go separate Decklink card, because usually we, in our productions, have to give out SDI signal, no need for another converter in that case (cheap option would be DeckLink Mini Monitor, it even has both, HDMI and SDI output - only one signal can be outputted), but really I'd go for DeckLink Duo 2, if your CPU has enough free lanes. You get 4 ports which can be inputs or outputs. 1st port for PGM, 2nd for GFX.