Probably the 6th gen processors were about the last to have WIN 7 support.
The first support to go was likely the intel integrated graphics which would not apply to you.
In terms of processors, the i3-10100 is a very good gaming cpu.
On a par with the r5-3600 and i5-9400, but cheaper.
But, if your cpu budget is in the $200 range, I would look at the 11th gen i5-11400.
Here is a review which includes 3600 and 10100 for comparison:
The Core i5-11400F coupled with a B560 motherboard make for an interesting build configuration, as it's significantly cheaper than the Core i5-11600K/Z590 and less than the cost...
www.techspot.com
If you have a win 7 license, give it a try.
If you have a problem, win 10 is quite stable by now and you should be able to activate.