The newer Intel 8th and 9th Generation processors are true quad-cores and they're decent budget processors for gaming, just as long as the system has a decent graphics card. Anything from the 1st to the 7th Generation is a dual-core with Hyperthreading, and the Operating system will recognize the older i3 processors as having 4 logical cores. Unfortunately for you the Core i3 has a bad word of mouth reputation which will make selling that PC a hassle because the i3 is not known for offering an optimal Gaming Experience. Today, a good 6-core CPU is needed for Gaming at the highest settings.
The GTX 770 was high end back in 2013 but modern games have exceeded it's capabilities. The most demanding of the Modern Games need a Graphics card with 4gb of Video RAM which puts the 2gb version of the GTX 770 at even more of a disadvantage. The GTX 770 isn't hopeless because you can still play modern games with it at lower screen resolutions.
I don't see you getting much out of your used PC. On craigslist I've seen systems with an older Core i5 and a GTX 770, GTX 680, R9 280x, or 7970 going for less than $300.
One thing that's overlooked is your expectations and needs for a new gaming PC. A lot of this revolves around the monitor you're going to use because that's the most overlooked bottleneck. What's the point of having an RTX 2070 and a nice Core i7 or Ryzen 7 CPU if you're going to play games on a standard 60hz-75hz 1080p display? You'd need at least a 1080p 144hz monitor to make good use of a setup like that.
Personally I would recommend looking at upgrading that PC with used parts if your fps goal is going to be between 60fps-75fps. The used or should I say the 2nd hand market is still flooded with capable cards like the RX 570, RX 580 and the 6gb GTX 1060. ebay is also flooded with used and cheap i7 and Xeon E3-1200 series processors which might be compatible with your system. Processors like the Core i7 3770 and 4770 are still good for gaming in 2019. You will have to figure out which processors are compatible with your system before making the purchase. Since you already have a GTX 770, you should be good with the power supply requirement.