It takes around a hundred thousand years for a photon from a nuclear reaction at the centre of the sun to make it to the surface. Effectively a diffusion style random walk in a highly scattering plasma medium.
Sunspots merely tweak the effective transport properties of the relatively shallow uppermost surface layer slightly.
Although it is true that the *sunspots* are cooler than the main photosphere there is one crucial point you are missing. The sun on average is mostly *brighter* when there are lots sunspots visible as the lost output from the spots themselves is more than compensated for by the much larger areas of bright faculae that accompany them. An active sun is a brighter sun this is not in dispute and is included in all the climate models. The effect of the sunspot cycle variation in TSI of 0.1% on the global climate is however right at the limits of detection.
You cannot blame the sun for all the recent warming - the satellite data rules out magically making the sun brighter.
Regards, Martin Brown