I have two of these same MB's, both are running great, one with an FX-6300, another with FX-8350. One's over two years old & only issue is [slightly] wobbly SATA ports, have swapped lots of drives. Even ran the Phenom ll X4 965 w/out issues for months, where it'll be going back into, need to move the FX-8350 back into the ASUS 970 PRO/Gaming AURA.
OK, I see what RAM you're running, do you have an extra set just for testing?
As Calvin7 replied, off the wall brand PSU's (I've never heard of the Aerocool brand) can make things go wrong. Speaking of the PSU, the two 4 pins plugs that goes into the MB, each is shaped slightly different, although can be forced in with ease (which will often damage OEM MB's with only a single 4 pin socket). You may want to remove those plugs, inspect each & make double sure that the correct shape goes in the right socket. This is important.
I've not had PSU issues because only run EVGA models, usually 650 Gold rated, although one of these same MB's has a B3 550W Bronze (Fully Modular), designed for small cases, as well as replacements for weak OEM units. Often needed if GPU is upgraded or added. Note that there's a Gold rated of the same size, being I trust the EVGA brand, has a 5 year warranty, neither is overloaded, these PSU's should run fine. Another is in a fully loaded XPS 8700 with i7-4770 & GTX 1060 SSC (6GB GDDR5) GPU.
As Calvin7 also stated, this was the only MB of this size with the 970 chipset available, the rest were older, some having SATA-2 support only, as well as odd mixtures of choosing between DDR2 & DDR3 RAM (two slots of each) & can only use one type, not both. Plus even with Full ATX MB's of this type, pricing has gone through the sky after having so many promos in the early Ryzen era. There's a few CPU's at decent pricing left (the FX-8350), the FX-6300 (at one time going as low as $69.99) is kind of pricey now, making the 8350 the better overall deal, a lot more CPU power for the dollar.
I suggest to check those two pins coming from the PSU, double checking to ensure that both are a perfect fit, use a magnifying glass if needed. If that doesn't get things going, if you have an extra set of RAM, try that & see if it helps. Not all RAM is compatible with all MB's, although to be fair, after so long, the MB OEM's doesn't keep an updated list. Bottom line, for a mATX MB, you won't do any better for the 970 chipset, if you don't want it, shoot me a PM, I'll take your troubles from your hands.;-)
Make sure the latest UEFI (or BIOS) update is installed. Should be 1.60.
http://www.asrock.com/mb/AMD/970M%20Pro3/#BIOS
Finally, a last check to ensure that all wiring is as should be. We all make mistakes, one wrong wire in the wrong place, and it messes things up. Also, the system may need to be reset (after BIOS update is performed). It can be done by removing the CMOS battery, press & hold the power button for no less than 30 seconds, and allow to sit for no less than 30 minutes. Then reinstall battery & power cord, fire up the PC & see what you have. Some settings will have to be redone, just want to get you back to the starting point, which should clear any incompatible settings.
If none of the above works, then try another PSU, of reputable brand. 600W isn't nothing if a knockoff brand, in fact, the PSU should be viewed as the most important component of the build. Why do I say that? Because inferior PSU's can blow away hundreds (or thousands) of dollars of otherwise good hardware. Bad ones, when they take a dive, not having the protections as the better brands, will wipe out a lot, the MB, GPU, SSD/HDD, most anything attached. In a worst case scenario, can also cause a fire. Even a good name Bronze, as long as it's rated for more than needed (some breathing room) are far better than generic branded models. Some of these may have been originally intended to be a name brand, the batch (yet to have any brand stickers installed) may not had passed tests during random checks & they sell to the highest bidder, who places their label on these & sells for a nice profit.
This way, the OEM breaks even and doesn't have to pay out of pocket for disposal. As many as most sells, can afford to unload a bad batch here & there.
https://www.kitguru.net/components/power-supplies/zardon/aerocool-integrator-600-watt-psu-review/2/
Note that on this 2nd page on down a bit, beneath 3rd picture, it clearly says mainly due to packaging (a thin strip of bubble wrap & thin foam on one side), this is a dead giveaway of a low end PSU. Yes the cables are sleeved, however the reviewer's sample didn't as much as include screws nor instruction manual (maybe your had both). Still, the packaging is a huge indicator of what one gets. It costs a lot a cash, as well as R&D to produce high quality products, therefore are packaged to protect.
On the last page of review, it says that the PSU should hold up fine'for a couple of years'. Lots of low end components inside if the full review is read.
Finally, this is around a 2014 model PSU, which if a good or top tier brand, may not be an issue, but for a low end model that's likely been in more than one build (the ASRock 970M Pro3 was produced & release almost 2 years later), that's an old PSU. No matter what MB you may replace the 970M Pro3 with, may have the same or worse issues.
Just my two cents worth.
Good Luck,
Cat