My recommendation would be to return the card and buy a card with the 8 and 6 pin connectors. The FE cards with just one pin are thermally throttling. There is a workaround that involves significant increase in fan speed (and noise) but you end up with either decreased performance or more noise, neither which I would consider acceptable.
Here's a comparison of the FE 1080 and non-reference 1080 from MSI
http://videocardz.com/60838/msi-geforce-gtx-1080-gaming-x-is-much-better-than-founders-edition
Image here:
http://cdn.videocardz.com/1/2016/06/GTX-1080-FE-clocks-over-time.png
The orange line (1670 - 1790) is the FE card with clocks drastically reduced by thermal throttling. The blue line is the non-reference card at a pancake flat 1910
NVIDIA’s own reference design suffers from severe throttling just after few minutes. It probably wouldn’t be that bad if not the frequency spikes. While average clock is somewhere around officially stated boost clock, those spikes cause micro-stuttering, which negatively affects gaming experience.
Hardware.info:
Founders Edition suffers from a horrendous amount of throttling and it runs +- 150 MHz lower all the time.
Meanwhile, MSI GTX 1080 GAMING X generates almost a straight line for GPU frequency (~1910 MHz), with no spikes and rather constant sub-70 C temperature. This should mean that the gaming experience will be much better, and card should theoretically generate better results in most tests. Also according to H.I. this is also the best custom design they so far tested.
The reference RX 480also suffers from a problem in this regard in that it overdraws power thru the 6 pin cable and the PCI slot which substantially exceeds the cable and PCI slot rating, users have already reported fried MoBos from this.
As for the 1070.... the FE cards are again throttling
https://www.techpowerup.com/reviews/NVIDIA/GeForce_GTX_1070/28.html
During gaming, the [1070 FE] card goes above 82°C, which results in lower clocks due to Boost 3.0.
https://www.techpowerup.com/reviews/MSI/GTX_1070_Gaming_X/28.html
During gaming, the [MSI 1070 Gaming] card also runs much cooler than the reference design, which avoids clock throttling above 82°C.
MSI GTX 1070 Gaming X = 52°C Idle / 71°C Load / 30 dBA Load
NVIDIA GTX 1070 FE = 39°C Idel / 83°C Load / 36 dBA Load
The non-reference card gets you 12 C lower temps, 6 less dbA, no throttling, no stuttering and higher fps.
As far as being safe... you will not that, unlike the 480, the 1070 FE stays within it's power limit with a 145 watt average load (Slot provides 75 w and the 6 pin provides 75 w for a total of 150w). Even at peak gaming, the card only draws 151 so you're not going to kill anything with your PSU / card combo.
Gaming PSUs are generally provided with 6+2 connectors so that they may be used on a variety of cards/ The most common at the high end is 8 + 6 on the card ... in which case you use a 6+2 for the 8 pin and just the 6 of the 2nd 6+2 and leave the 2 dangling.