It should hold up, but will have some bottlenecking. Here is a 1080 comparison and you can see some games get 20 to 30% less fps, but it's still performing very well. If you go higher than 1080p there is almost no bottleneck.
https://www.guru3d.com/articles-pages/amd-ryzen-5-1600-review,19.html
You really should get a better cooler. You may wa t to look into overclocking, it's simple with these.
If you are only playing in 1080p you m8ght want to check the 1070ti benchmarks and see how far it varies from a 1080ti. For example if a 1080ti reaches 150fps in game X with a 8700k and with a 1600 you hit 120fps, maybe a 1070ti hits 120fps with the 1600.. but if you are going higher than 1080 it seems like it should hold up well enough..
Had some free time and did some searching. It's tough to find better comparisons.. at 1080p a gtx1080 on tomb raider pulls in max 140fps and thr 1600 hits 112.. the 1080ti same game has max 170fps and the 1600 hits 130..
This is a rough comparison, not even sure the settings are 100% the same, but in this one it's probably better to just have a 1070ti for that cpu with a difference of 28fps vs 40fps when using the best cpu available. Also the second one used newer cpu for the max fps..
You can see though like I mentioned once the resolution goes up the difference is much smaller
https://us.hardware.info/reviews/8267/12/amd-ryzen-5-2600-review-the-most-interesting-ryzen-2-gaming-benchmarks-rise-of-the-tomb-raider-dx12
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