As I understand it our modern notion of what rigorous mathematics is didn’t exist back then. The justification for analysis was basically physical intuition and

As I understand it our modern notion of what rigorous mathematics is didn’t exist back then. The justification for analysis was basically physical intuition and simply that it worked. Euler and company just worked directly with the intuition of infinitesimal quantities. To be fair though, infinitesimals are the essential intuition for how calculus works anyway. Personally I don’t really care about constructions of the real numbers or technical details of calculus. It’s enough for rigorous mathematics to know that models exist of the theories which produce calculus; that is models of complete ordered fields and even plain old ordered fields with infinitesimals.