Cell voltage doesn't tell you much about whether or not a cell has desulfated. You need to measure the specific gravity of the electrolyte using a hydrometer to tell if the sulfur deposits have been restored to sulfuric acid and that the cell has taken more than a surface charge.
Specific gravity is temperature specific. A good hydrometer will have a calibrated float, the best will also have a thermometer so you can compensate for the electrolyte temperature. The multi-ball type of hydrometers are about useless.
An equalizing charge serves two purposes. First, since all of the cells are in series, they all get the same amount of charging current. If one cell is weaker than the rest, it will charge to it's rated voltage first, and the battery will indicate full voltage before the other cells reach full charge. The higher equalizing voltage forces the current flow to continue, letting all of the cells reach 100% charge.
Second, the bubbling mixes the electrolyte, reversing the stratification layers that can form. Sulphuric acid is heavier than water, so without periodic equalization it tends to settle towards the bottom of the cells, leaving inactive water at the top.