Personally for those, I switched my English around instead.
“Because I am weak, I go to the gym. "
" I am weak, so I go to the gym”
“Because I don’t like vegetables, I don’t eat them”
" I don’t like vegetables so I don’t eat them"
I don’t see a huge problem with your interpretation of it, because honestly the English you translate it to in your head/on paper to make sense of it is just a foothold for now. As you learn new grammar and gain experience with the old, your “translations” or feelings/sense of the grammar points will shift and adjust to make room for more to place next to them. So for now sure let them be “that’s why” but don’t be surprised if that gets more elaborate/ changes slightly as you gain experience and comparable vocab/grammar.
But also remember you are learning Japanese , which wasn’t made for the purpose of being closely translated to English. It’s got it’s own thing going on so let it be free and embrace it ฅ^•ﻌ•^ฅ