The FIRST thing you should have done, and let me just say congratulations on being wise enough to choose an EVGA graphics card if you're going to go Nvidia, but really you are in serious need of a power supply at if you were seeing issues with a lack of performance on your GTX 970 there's a very good chance it was due to inadequate power delivery.
And even if you were NOT seeing that, and were just feeling the itch and a desire for a bump up to something newer and better, let me just say that you would be VERY wise to not run that new graphics card on ANY power supply that old, especially when it's a unit that only had a five year warranty so is clearly long past it's intended life expectancy.
Truthfully, that unit is
probably more like 10-14 years old since ALL the reviews of that Silencer 610 EPS 12v unit are from back around 2008 and I know they didn't continue making that model for more than about two or three years, at most, before it was superceded by the Silencer MKII in 2010. It HAS to be at least 9 years old and that's about four years more than it should have be trusted to continue being in use for with any hardware you value and would like to see stick around. Literally, being as old as it is, whether it actually currently has any obvious faults or not, it should be discarded because it is definitively at the age where capacitor failure is no longer an "if" but a "when". As are likely some of the components that make up the protections in that unit. It could die with a whimper but it could just as well be more of a hammer to the forehead of your graphics card or motherboard or simply a potential fire unto itself that lets out some magic smoke and gets rid of some of that unnecessary plastic insulation on those pesky cables.
Seriously though, definitely not worth taking ANY kind of risk with a brand new 500+ dollar investment, when a very good, safe, replacement unit can be had for around 80-100 bucks.
I would absolutely not use that card until you replace that power supply. A GOOD 550-650w unit should be plenty for that or any system using that card, if you are not doing any significant overclocking of any kind. Might want a 650-750w unit if you do plan to do so.
Below you will find MY standard list of recommended power supplies and beyond that this thread is intended as a landing place for questions or discussions regarding specific units, platforms or related PSU tech, all of which are all welcome to be discussed here. If it's related to power supplies...
forums.tomshardware.com
As far as the platform goes, I'd have to agree that you should not be seeing any GPU limitations at 1080p with the GTX 970. The 2070 Super is more like a 1440p card, but you certainly won't be GPU limited at 1080p no matter what game you are playing with it. Probably, any limitations you have are FPS limitations and those are going to be due to your fairly old, low core count i5. Practically ANY current mainstream Ryzen or Intel CPU will be a good choice by comparison.