1) What is the type of your existing internet service? You say not Fios but does that mean Cable or DSL? What maximum advertised speed? (And what speed do you get off a strong-connection device going to something like
www.speedtest.net ? What kind of numbers in the weaker part of the house over Wifi?)
2) What is the make and model of the modem+router device you have now from Frontier?
More than likely Frontier is using MoCA outputs from the modem+router to service the cable box. Assuming I am correct, it will be a pain to remove the Frontier-supplied modem+router box completely, even for someone pretty technically skilled it could be something of a project.
You can more than likely use a powerline extender pair directly off of the existing Frontier router+modem box, assuming you have a spare ethernet port on it. You'll want one of the ones that has a Wifi extender integrated given what you described, but ideally your sons gaming computer can hook up to the ethernet port on the far end device. Some more info about powerline networking is
here but don't buy more speed than you need (not much point in having in-house speed greater than what you are getting from Frontier, thus question #1 above).
You can also go for the Google/Nest Wifi but you'll be better off putting that into some kind of bridging mode, rather than having stacked/chained routing which can make a mess of some things. (Again, possibly above your troglodyte threshold.)
If your house is really, really old (?) the wiring can affect the quality of the powerline networking - I don't have a lot of experience on how bad that really is, since I've always lived in houses built within the last 30 years. I'm also assuming you keep your basement at least above freezing at all times, since some gear (probably including the Frontier modem+router box doesn't like things that cold).
Assuming you have something like 100-150Mbps service, something like
this kit for $48 (prime eligible) would probably do the trick, even with old wiring it might be an upgrade over the issues you are having with plaster + wifi right now. You might also need to buy 2 ethernet cables, unless you have some extras lying around. One needs to reach from the Frontier modem+router to the closest AC outlet. The other needs to reach from the closest AC outlet in your sons' room to his gaming PC.
Note: sometimes the powerline adapters cover both outlets, depending on exactly the style and placement of the AC outlets you have. Some of the "fancier" ones have a pass-through outlet because of this, but they are usually more expensive. In any case, you aren't supposed to connect them via a power strip (but if you get the pass-through, you can plug the power strip into that.)