The Core i5-3570 performs significantly better. The Core i3-6100 only has secondary advantages, such as lower power consumption, newer platform features, and a better upgrade path.
You'd need DDR4 RAM with the i3-6100. Don't get a Skylake DDR3 motherboard as those seemingly have safety concerns that could fry the motherboard. So clearly the i5-3570in this situation.
Performance of both is quite close, but the 6100 comes with DDR4, not to mention it's actually new and not 4 years old, so if it was me, definitely for the i3-6100. Warranty and stuff.
The Core i5-3570 performs significantly better. The Core i3-6100 only has secondary advantages, such as lower power consumption, newer platform features, and a better upgrade path.
I'd say the i5. Frankly, Intel hasn't improved performance that much since Sandy Bridge, and for cheaper than the i3 you get a chip with four actual cores.
You not OC'ing the i3-6100 with a 50-60 dollar mobo like OP mentioned though. The i5-3570 is the actual cheaper option for him. So i5-3570 is the only logical option but I'll leave it at that.