The gear stick side to side movement translates to up and down movement of the selector shaft sticking out the top of the gearbox. This is what the bellcrank mechanism operates, with the "Top Hat" bushes used to support the bellcrank shaft. As the alloy support bracket corrodes, specifically where the bushes fit inside it, the corrosion compresses the bushes against the bellcrank shaft ... causing it to stick. Getting rid of the corrosion is the important thing so the shaft is free to rotate. Greasing it up just helps to prevent further corrosion.
The gearstick fore aft movement translates to rotation of the gearbox selector shaft. Since the shaft is directly connected by a simple lever arm to the shift cable there is nothing to stick so this movement should not be restricted by the mechanism on top of the gearbox. It's possible the cable itself might be sticking but this would affect all gears.
No harm in cleaning up the selector mechanism but I don't think it will solve stiff selection problems for the gears you mention.