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.
You are technically right that the water heater softens the water a bit by precipitating the minerals around the heating element and thereby removing them from the water. But that is energy inefficient and expensive, since you normally don't use a water heater to soften your water but rather to get warm water. So putting another system in front of the heater that softens it first is better than replacing the heat element every so often.
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?
Maybe with that attitude. You are willing to swap a heating element but not running a gas line? It ain't shit but playing with black pipe and dope like a white girl with daddy issues.