Can the fact that energy is distributed homogeneously in the universe be explained through heat flow and not the Cosmic Microwave Background(or in other words, can we say CMB is a result of heat flow and not the universe expanding from a singularity)?
The evidence for the expansion of space does not come from the Cosmic Microwave Background. It comes from measuring the velocities of receding galaxies and clusters of galaxies, the velocity measured byt the doppler effect found in spectra of specific atoms from distant galaxies.
The cosmic microwave background was a prediction of the expanding model of the universe and was found by observation later.
The homogeneity of the distribution is another story, and would be unexpected if the universe developed for 13+ billion years randomly, as in the beginning there was not thermodynamic possibility of uniformity in the energy spread in the Big Bang model: due to relativity not all regions could exchange energy with all regions so as to attain thermodynamically uniform distributions. Within the BB model this forced the inflation period in the beginning and postulating quantization of gravity and the inflaton making the distributions uniform.