The independence has nothing do with the integral being 1 to be honest. You could write a model where the observations are not independent but the (multivariate) integral over their domain will still be 1.
If by “joint probability” you mean function(params, data) there is no joint probability here in general.
L(params, data) is constructed from a family density functions p(data) for each possible value of param. The integral of L(params, data) may be anything or diverge. You don’t need any extra independence assumption either.
Or maybe you mean “joint probability” as p(data1, data2) when data is composed of two observations, for example. But you don’t need any independence assumption for that probability density to integrate to one! It necessarily does that - whether you can factorize it as p’(data1)p’’(data2) or not.
