being outside of that 10% rating will not necessarily cause immediate issues, but it will cause problems over time via a shortened lifespan on parts, or the power supply itself.
The power grid is meant to run things such as fans, engines, and compressors that are found in your fridge/HVAC/washer/drier/etc, and things that heat up like old edison lighbulbs and heating elements. These things all have a high current to start, but then level off for a long relatively low, but consistent power usage.
Computers on the other hand will run at 2-5W in sleep mode, 25-100W in active idle, 75-150W under normal use, and then jump up to 500+W when doing major work such as playing computer games. And the whole time that wattage is variable, instead of running at a singular consistant power consumption rate. It's not that it is a bad thing, but the power grid in older areas is simply not made for it. It is such a problem that some new homes will have a capacitor built into the home to help alleviate some of these fluctuations, and newer negiborhoods have a great many more capacitors and transformers on the telephone poles compared to older ones, all to help average out the load more over time for the power stations.
Living in a newer neighborhood (you said your home is only ~5years old) it should not be an issue because current electrical code is set to be able to take the hit of these newer devices. But older negiborhoods like mine are ~60-70 years old. In fact, my home is so old that it originally had ~5 lights, and a grand total of 11 outlets, and no ground. But the point is that while I now have a modern electrical box, grounding and GFCI in most of the house (excluding the original circuit), and much higher amperage service than it had originally, the local electrical grid was simply not meant to take the kind of load I (and my neighbors) put on it with our computers, TVs and consoles. Plugging a gaming rig into an outlet without a battery backup around here is simply suicide. Even with a decent power supply that will clean up a lot of the local power issues, the voltages in my rig are well below where they ought to be when pushing the system. But you throw it on a battery backup that prefilters and buffers the power flow and my rig runs within ~5% of what it is rated, which is great!
So, all that to say: Yes, a battery backup, or a power strip that does a bit of filtering (something more advanced than a simple breaker) can make a huge difference, and is a great precaution to have in any area prone to high winds, electrical storms, power outages, or generally sub-par power situations.
No, having your voltages off even as much as 15% (you are at 12.75%) is not going to necessarily cause stability issues in your games, because electronics makers know that most households are not in 'ideal' power situations and design some fault tolerance into their hardware. But it is going to put more strain on your hardware, which will lead to shortened lifespans, and can cause odd issues over time.
So it can help, but it does not rule out the possibility that you simply got a bad power supply. But regardless of your power supply, you should have some form of filtering, or battery back up in place 'just in case'