No, but even a slower Core 2 Duo should be hitting the limits of a 8600GT on regular basis. Afterburner can confirm, I think, and GPU-Z, probably, if the OP wants to check, but hasn't.
I don't think getting a decent GPU now, with a lower budget, and then a good CPU upgrade later, is a bad way to go, especially considering that the OP is not too concerned with IQ. Upgrading the CPU but not GPU, or using IGP (with AMD), would still be more expensive, even selling old parts, unless the OP is using a retail Windows, and would still result in quite a compromise, v. having a 6-moth-old, say, GTX 750-ish card, with a better CPU. At lower settings, GPU-limited cases should remain fairly rare, for at least a couple years, in the kinds of games the OP is mentioning, with something in the performance range of a GTX 750 (Ti).