I recently came across this: 家族のありがたさをつくづくと感じる
On first inspection I presumed it meant that he was feeling how grateful his family was of him however the text was translated as:
家族のありがたさをつくづくと感じる - I feel deeply grateful to my family
How can you tell whether it was the family's gratefulness or person/author? Thanks
Okay, I got a bit too curious for my own good and looked it up. I think the assumption in my comment was correct. 有難さ doesn't actually mean gratefulness according to kotobank. It actually means "something to be grateful for" or "value". I found two examples where it actually expresses this. The first is an example I found on tatoeba.
水不足の時、水のありがたさを骨身にしみて感じた。
During the water shortage, the value of water really came home to me.
Another one I found was a sentence on a hinative post comparing 有難み and 有難さ. They essentially mean the same thing.
大人になって、初めて親のありがたさがわかる。
Once I became an adult, I finally realized (how much my parents did for me / that I should be grateful for my parents).
If you think about it, this makes sense. 有難い doesn't actually describe a grateful person, it expresses that you are grateful for what it describes. For example, 有難いこと means "something that (we) should be grateful for. You wouldn't say 私は有難い.
So, I think the original translation of your sentence was correct. I hope this clears things up. Also, if I got something wrong, please correct me.
Edit: Here's another example I found on Weblio
あなたのそのお気持ちは大変有難い。
I am grateful for your feelings.
