Why should MLE be considered in Logistic Regression when it cannot give a definite solution?

I think you are comparing apples and oranges here. Maximum likelihood is a the maximum value of your likelihood function, which somehow describes your data generation process. Specifically likelihood gives you the probability of observing your data, given the data-generation model you imagine. It is similar to a loss metric in that respect.

Gradient descent is an approach to varying your parameters in such a way as to maximize/minimize some function, e.g. loss-metric.

So why are you trying to compare these two things? It would seem to me that you can use likelihood as a loss-function (normally log-likelihood) and then run gradient descent to maximize it.

Perhaps that's what you meant. Why would you use likelihood as the loss-metric for gradient descent? In my use-cases it helped when available data was not evenly sampled or non-homoschedastic, e.g. if you want to do regression to get y as a function of x=0...1, but variance of y is greater in the region x=0.2...0.4 compared to other regions. Using least squares as the loss metric may give a poor fit (since the intrinsic assumption of least squares is that your variance in y is the same).