How remove sash bracket and spiral balances

You have what's called builder's grade windows, which were mass produced in the mid 20th century and were explicitly designed to not be repaired. Instead, they would have preferred if you bought an all new window.

That doesn't mean that you can't repair this window, just that it won't be easy. After all, you're trying to do something that the makers of the window specifically didn't want you to do. To remove the spiral balance, you need to detach it at the bottom (making sure to keep a solid hold of the sash so it doesn't fall down), detach it at the top, and then do your best to pull it up and out from behind the sides of the sash.

This could be trivial or difficult depending on how large the top sash is relative to the bottom. If it's the same size or larger, you should have enough space to pull the spiral balance out without much fuss. If it's smaller, then it will be extremely difficult to get the right angle for removal. You just have to try your best in that circumstance.

The only other way is to completely disassembly the frame of the window, which is a truly massive job and not one I would recommend you take on lightly.

Sure, to detach the balance from the bottom corners of the sash I would strongly recommend using a 75-005 tool. It will make your job a lot easier since it's made to do this exact thing. What you need to do is stick the tool into the gap from underneath and hook it into the hole at the end of the spiral rod. Then, just firmly pull down. That will disconnect the spiral from the clip it's attached to. To install the new balance, you just reverse the process.

To clarify, there is a bracket in your window. I can see it in the second and third photos above. It's just installed on the side, obscuring it from view and convenient access. Remember, the manufacturer didn't want to make this easy for you. You have to fight against their design.