For continuous data it is exactly a probability density evaluated on your data (for discrete it's PMF instead).
L(params)=P(D|params)
L(params)=P(D|params)
So we ARE working with a probability function. Its output can be interpreted as probabilities. It's just that we're maximizing L = P(events | params) with respect to params.
There is another function - a function of data for fixed params - which is a probability density. That doesn’t change the fact that the likelihood function isn’t.
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.