Even then you'll need 27 months (2 years and 3 months) before being more cost effective than Geforce Now. That's considering that you are on a desktop though with a CPU powerful enough.
Personally I started doing cloud gaming mostly because I was on a netbook at school (I liked the form factor and the battery life was quite nice versus most laptop). At first on OnLive, then on LiquidSky, and since a year ago, on Geforce Now.
I don't believe the service will stay at 5$/month though, both service I used shutdown because they weren't viable and they were in the 10$ range. LiquidSky was supposed to come back, but they missed multiple release date now and with the release of GeForce Now, I have no hope. Shadow seems more viable at 25$/month, which makes your solution much more cost effective.
You can pick up a used i5-3xxx or 4xxx system with 8GB of RAM real cheap. If you get one that has at least a 300W power supply, you can then put in a GTX 1660 or similar and do quite well with most games. Max settings @ 1080p for some older games, and still decently playable at decent quality for the newer ones. And all for $300 USD or less.
I just put an RX 580 with 8GB of video RAM into my old i5-3570S system. The card was just $135 at Microcenter, because it was open-box.