First of all, I did a quick check for reviews about that silverstone psu(I'm not personally
familiar with the brand) and found a very positive review on hardwaresecrets.
http://www.hardwaresecrets.com/article/SilverStone-Strider-ST50F-500-W-Power-Supply-Review/548/1
They know their stuff as far as running a PSU through the wringer goes, so I trust them.
Just be careful about which components you connect to certain power leads so that
the pull on the split +12v rail is balanced.
That 7850 is twice as much as the 7770(250au vs. 125au), but is not even twice as fast.
However, after analyzing your intended uses, that particular 7770 may not be the best fit
for you. Per wikipedia, LoL is not very demanding, and I imagine any indie game would
likely be even less so. For competitive shooters?(willing to sacrifice detail for higher fps),
you want a gpu that can handle the one toughest on systems(BF3).
The linked 7770ghz Ed. has a reference styled cooler with reference 1ghz core clock, so
you may not be able to push it to 1.2 without adding voltage. The linked 7850 has a
custom dual fan cooler and ships at 975 core vs 860. The catalyst driver might limit higher
overclocks to 1050, but something like MSI Afterburner(free download) will let you go
higher and adjust voltage(both would require maxing the powertune slider in catalyst).
I don't know much about 7850 OCing, but I don't think you'd get much over 1200mhz at
best. A 7770 with the right cooler can push 1250 on the core. One like this:
http://www.centrecom.com.au/catalog/product_info.php?sort=3a&products_id=56574
A good sized fan and copper heatpipes allowed some reviewers to get well over 1200
while remaining cool and quiet.
When the 7770 first launched, hardocp tested a somewhat overclocked dual fan xfx
version against both hd6870 and gtx560ti and found it delivered the same gameplay
experience(same settings at subjectively playable frame rates). Here is that review:
http://hardocp.com/article/2012/02/14/xfx_r7770_black_edition_super_overclocked_review/1
Pay particular attention to the BF3 page. They also overclocked it to almost 1200 easily
and without a voltage bump. The sapphire is and even better card. Plus, both 7850 and
7770 get a speed boost by recent catalyst drivers that use the new GCN architecture more
efficiently. The sapphire costs a bit more than the other 7770, but is still a better value for
your dollar than the linked 7850 and will let you do what you want to do very well while
remaining within the confines of your budget. (based on mhz x functional units on die).
http://en.wikipedia.org/wiki/Comparison_of_AMD_graphics_processing_units#Southern_Islands_.28HD_7xxx.29_series
I don't recall whether you said you'd be interested and/or willing to overclock to get
the best bang for your buck. Both the 7770 and 7850(28nm process) beg to be
tweaked. If you do a little research and are careful, you can achieve a safe and
effective overclock. Whether you do or don't, the sapphire is still a better value for the
money vs that 7850, factory clocked or hand tuned. You may not be able to game
smoothly at 1080p ultra +fxaa(minus motion blur for both performance reasons and
competitive edge) with only ambient occlusion bumped down a notch from horizon based
to screen space, but BF3 still looks pretty awesome on high(esp. compared to consoles).
If you want to save a bit, you could go with the xfx 7770 and play BF3 on high/w fxaa.
Yes your preference is max settings, but only if the budget permits. High is better than
medium("average").