At first level, some classes get more starting funds than others. But after that, everybody uses the same WBL chart. Why?
Obviously there is some reason for those classes starting off with different funds. It seems that it was done for balance reasons so that classes like Fighters can buy expensive armor. Shouldn't it continue to scale after first level to be fair?
The starting cash is half flavor, half for the being able to purchase expected starting gear. For example, a fighter will need a few weapons, armor, and travelling gear. A monk doesn't need armor or weapons, so just has to get some travelling gear. Past 1st level, the charcater are looking at costly magic items and the low price of mundane gear isn't important anymore.
If you were in an adventuring group and played a class with less need for gear, would you give the fighter more treasure than everyone else or would you expect an equal share for yourself? A group of players wil generally go for more or less equal division of treasure, with certain items going to the character who can best use it. But that would be on a case by case basis.
Some classes do get more WBL than others. Base Alchemists and Wizards for example as they have item creation feats baked into their class. I believe the standard is +25%-50%? Although that must be in items that you could have crafted, so scrolls and potions respectively.
The game is built on the idea that all characters start at first level, and the starting wealth does help balance the characters with equipment vs skill and abilities. Once they start adventuring all those first level characters are theoretically equally sharing the same loot. So the fighter and the wizard and the rouge all gain about the same amount of wealth and still have their perspective amounts of starting gear. The wealth by level helps DM account for the approximate amount of funds an adventuring party should have to be balanced with the CR's they will be experiencing at those levels. There isn't a need to give more loot to one class over another and if you are starting characters at a higher level they all gain the same amount of loot plus their starting gear so the balance is maintained.
At first level, some classes get more starting funds than others. But after that, everybody uses the same WBL chart. Why?
Obviously there is some reason for those classes starting off with different funds. It seems that it was done for balance reasons so that classes like Fighters can buy expensive armor. Shouldn't it continue to scale after first level to be fair?
Once you go beyond the basic equipement, the costs basically equal out. It might cost more to have a set of breastplate then studded leather, but the difference between +3 studded leather and +3 breastplate is far less significant relatively speaking.
Also the assumption is that a party splits the wealth gained evenly as they progress. which would obviously put all the party members roughly even in wealth long term.