The sacrificial anode is there to protect the steel tank. It lasts a long time.
This is a hard water problem as everyone else is saying, and a water softener would solve the issue.
*Edit: check the very bottom of your tank since you have the elements out. It most likely has a pile of calcium and other minerals sitting on the bottom.
You can’t filter out ions of calcium like that. A huge reverse osmosis system for the entire home would be prohibitively expensive. I used to live in an area with very hard water and everyone had water softeners. You only need to buy the salt every few months and it’s not too bad. RO filters were only connected to a tap on the side of the sink in the kitchen - those membranes aren’t cheap.
I'm guessing the inside of your tank looks just like this and swapping new heating elements in isn't going to fix that. Maybe try flushing it out first?