That is just my opinion.
Here's the deal. The i5 will be a little faster for games today vs the Ryzen 3. No question of that. Here's a comparison
Ryzen 2200g vs i5 7500
https://www.youtube.com/watch?v=bhffayvY7oo
However here's another of that Ryzen 3 vs a Pentium G4560, so the pentium may be slightly faster than yours.
https://www.youtube.com/watch?v=Z1T4TAaLxqQ
So the first one is the i5 vs the 2200g which is newer and probably cheaper than the 1200. That gives you an idea that yes the i5 is a bit faster. Overclocking the Ryzen will help to even that gap a little. Plus if you are playing at 1600x900 on a 60hz screen, you aren't really seeing over 60 fps anyway.
The 2nd video gives a baseline of where you are today vs the Ryzen cpu.
Basically, you can go and get the i5, no question, good upgrade for you, you'll run games better etc. If you ever want to upgrade beyond a 7700k however, you are looking at buying a new cpu, and new motherboard at least, as Coffee Lake doesn't support the kaby lake boards. So that can get pricey.
With the Ryzen setup, you are sacrificing a little performance right now, but it will likely do what you are wanting to do, and you said you would stay at the same price as the i5 you looked at. So with the Ryzen, it's sacrifice a little performance now, but later, then say Ryzen 2 or Ryzen 3 come out, since they are supposed to support socket AM4 until 2020, if all goes as they are saying, you should be able to upgrade from your Ryzen 3 to for example a Ryzen 7 with 8 cores 16 threads or whatever they have at the time that your motherboard will accept. Also, with Ryzen, keep in mind, AMD didn't have the greatest performance before. But Ryzen was about a 50% jump in performance vs their older chips. Being that we are still on the 1st generation Ryzen, I am assuming that they will refine the chips to get better single core perfomance and also have higher clock speeds at stock which will help.
So in a couple of years, when you upgrade to say a Ryzen 3rd generation 6 or 8 core etc, then you get a little bit of extra life out of your system, whereas the i5 is essentially the same performance as the new i3 chips.
In my mind, it's like this
More performance now=i5
Upgrade path for later with enough power for right now=amd
Again, maybe I'm wrong, but that's just how I see it. If you were going to sink money into the current platform you have, I would almost consider saving up and getting an i7 7700 or 7700k and be happy for a few years instead of getting the i5. But there again, more money. At that point however, you'd be better off upgrading your graphics card and monitor anyway once the prices go back to a sane level.
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