Your assessment is correct. Class balance and skill balance are the two major factors.
If we assume that the designers aimed to make balanced classes, then making a class better at something means it has to be worse at something else.
Skills are just one part of how classes are different, a class that is good at one aspect (combat? spell variety? hit die?) may not be as good at something else.
Skills are not equally valuable: Perception is probably more useful than Religion. Stealth is probably more useful than History.
Skill rarity is another balance concern. Heal and Endurance are much more common class skills than Dungeoneering.
Skills share attributes with more or less than others. There are only one Constitution skills, only one Strength, but 5 Wisdom skills. This means it is easier for a character with high Wisdom to be effective in many skills than a Strength character. This leads to even more complexity as changing the attribute of a skill changes how powerful it is.
I find it hard to give exact examples of X class has more skills than Y in exchange for being worse at Z. Classes vary so much in their roles and capabilities that identifying any exchange is difficult.