> However, the idea is that often a lot of the probability mass - an amount that is not small - will be concentrated around the maximum likelihood estimate, and so that's why it makes a good estimate, and worth using.

This is a Bayesian point of view. The other answers are more frequentist, pointing out that likelihood at a parameter theta is NOT the probability of theta being the true parameter (given data). So we can't and don't interpret it like a probability.