Solution 1:

Yes it's correct, because 'it' is the subject of the sentence, and it is singular. It's the same kind of 'it' as:

It was raining.
It was the reason why he couldn't bear to stay in the house.
It was the children who let the dog out, not the postman.

Made is the verb that belongs with vespertine saunters.

You could leave out the 'it' part and the sentence would still make sense:

Those vespertine saunters made him survive the work week.

"It was....that..." is there for emphasis. (It was the vespertine saunters and not anything else that made him survive the work week.)