How to get agitator off for this Frigidaire washer?

I have seen plenty of youtube videos on how to release the agitator from rest of washing machine. Seems easy.

However, for my FEX831FS4 washer the bolt is kinda surrounded by the agitator’s plastic housing and the only way I may be able to unscrew that nut would be with a very deep, thin walled socket. Wrench does not bite the nut enough and regular socket can’t get on top of nut because the protruding thread is very high. See this picture to understand what I mean

This has made me wondering If I am doing things the right way to fix the broken agitator? Wanted to hear from someone who has worked on washers before whether

1. I need to start disassembly from bottom for this washer and not top to release the agitator?
2. When the agitator got broken perhaps it was pushed upwards (or downwards) over bolt and I now simply need to use some force?
3. I simply need a special, deep socket with thin walls to unscrew that nut?

Any thoughts? Here is parts schematics. If I understand correctly then last picture of transmission shows that bolt. But then it seems that the plastic agitator should simply come after unscrewing the plastic cap without having me to unscrew this bolt. Which does not seem to be the case.