others here would convince them to buy an 8 year old GPU that is mostly going for ~$150-300 used.
I am sensing hostility. I understand it is an old card, however it is an option that has its pros.
As for the price, that varies region to region. In USA on ebay, you can get 2gb varients for well under $150.
Examples of some cheap ones:
PNY GTX 770 2GB $99+shipping (+$9 for me)
Msi TF 770 2gb $99+shipping (+20 for me)
Going with ebay would give the buyer protection so that you can be made sure the card will work fine.
So is the 5600G the best option?
There is no right or wrong choice. Both options have their pros and cons.
The 10100f or 3300x with a 770 would offer the best performance in games currently. While it still isn't a card for high settings, it will perform significantly better than the 5600g+igpu in games. It will be night and day difference in my opinion.
The 5600g and igpu will offer less graphics power, but on the flipside a more powerful CPU. This option also comes with all of the benefits that comes with that. Warranty, better driver support, etc.
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To note, if you are going the route with a used descrete GPU, I would not reccomend the 3300x.
If you can afford abetter cpu than the 10100f, I would go with an i5 10400f at $150 with same $105 h570 over swapping to a $140 3300x and $100 b550. For $15 more you are getting a significantly better processor with 6 cores versus 4. It does have downsides, but i think the pros outweigh the cons.
A 10100f, 3300x, and 10400f all hold their own as CPUs when paired with newer GPUs. You could easily upgrade the GPU to a much more powerful newer card without worrying too much about the CPU holding you back.