Yes, rewriting them that way would be correct.
Aside from the meaning of "Because", "아/어 사" Can be used in the following ways:
As a clause connector in which the second action occurs after the first (similar to 고). This is usually added to 오다 and 가다:
우리는 집에 와서 바로 잤어요: We came home and went to sleep immediately
저는 당근을 칼로 잘라서 냉장고에 넣었어요: I ...
되도록 by itself means "as much as possible", similar to 될 수 있는 대로, 될 수 있는 한, 가능한 한, 가급적이면, etc.
Your sentence can be interpreted in two ways.
[아무리 바빠도 되도록] 저녁 식사는 ... = We should have dinner with family so we can be as busy as possible (makes no sense).
아무리 바빠도 [되도록] 저녁 식사는 ... = No matter how busy we are, we should have dinner with family as much ...
I think it's misleading to ask when "아/어서" means "to" - they are both very versatile constructions that can be used in various situations. So, some situations can be nicely expressed with "아/어서", some others with "to", and some others with both.
In the examples here, "A to B" can be interpreted as "A in ...
The first “였다” is “이었다"'s abbreviations.
Noun: noun which has “받침” -이었다(었다)
noun which doesn't have “받침” -였다
verb: behind verb which is “past tense” (except for ㅏ, ㅓ) -었다
verb which attach(use)”~하다”,”하다” -였다