Why estimate using GEEs inspite of their disadvantages over using ML? I know sometimes I want to know the population-level estimates, but the problem with GEEs is I can't calculate the likelihood, and therefore all models I make with it aren't comparable, and I don't know which model is right. For all I know what I am fitting with GEEs could be the noise rather than the true value. If models are specified differently, the entire conclusions can be different. There's Quasi-likelihood but I can't compare the quasi-likelihood to a chi-square distribution and do likelihood ratio tests with it. 
Someone told me that GEEs are interpreted in the case of drug trials as if I were to give the entire population (all the people) many of whom are not sick, the drug, what kind of effect we would see. 
But that would be stupid unless we're measuring clinical phase 1 trials, testing the toxicity on a generally healthy population.
Under no case would you ever want to give the drug to the entire population( that's impossible and useless and never going to happen). So why use a model that estimates what would happen if we did give the treatment to everyone on earth, the universe, and beyond?
I disagree with the contention that no model is correct; I believe there is an objectively correct model. To say no model is better than the other is to misunderstand or reject the validity of maximum likelihood and all good modeling metrics. 
Why use GEEs?
I heard it's like the difference between public health and individual health. But why are the estimates of individual effects so different from population-level effects when individuals are supposed to be representative of the population level?
 A: The GEEs have been originally proposed for clustered/grouped categorical data. Namely, when we have clustered/grouped data, it is expected that measurements within the same clusters/groups are correlated. In the case of normal data, we can go from the univariate normal distribution to the multivariate one to account for these correlations using the variance-covariance matrix of the distribution.
However, in the case of categorical data, going from a univariate distribution (e.g., Binomial of Poisson) to a multivariate one is not that straightforward as in the normal case. Liang and Zeger proposed the GEEs as an alternative. They observed that in the standard generalized linear model, the score equations contain a diagonal matrix with elements the variances of the observations implied by the family. They argued that we could replace this diagonal matrix with a full matrix containing some working correlations. The theory they next developed suggested that solving these equations would give asymptotically unbiased estimates for your coefficients, but with standard errors calculated with the sandwich formula. Moreover, the use of the sandwich estimator for the estimation of the standard errors provides protection against misspecification of the working correlation structure.
However, there is no free lunch! In case you have missing data in your outcome variable, the GEE will give unbiased estimates only when you have missing completely at random (there are extensions, such as the inverse probability weighted GEEs that will work under missing at random, but require that you correctly specify the dropout model). Also, the sandwich estimator will better protect you for misspecification of the working correlation when you have relatively few covariates that are continuous, and you have balanced data. 
A: Why use GEE (instead of Maximum Likelihood, ML)? Because the likelihood could be wrong. In fact, since we never know if it's right or not, GEEs safeguard against biases. We want results that require few assumptions. GEE's assumptions are some of the most general, hence the Generalized "G" of GEE.

... the problem with GEEs is I can't calculate the likelihood and therefore all models I make with it aren't comparable and I don't know which model is right. 

You can absolutely compare nested models with the robust score or Wald tests. These results usually mostly agree with ML. Also, you never know if a model is right. Tests don't tell you that.

For all I know what I am fitting with GEEs could be the noise rather than the true value. 

True of ML as well.

If models are specified differently the entire conclusions can be different. 

True of ML as well.

Someone told me that GEEs are interpreted in the case of drug trials as if I were to give the entire population (all the people) many of whom are not sick, the drug, what kind of effect we would see.

This is a misunderstanding of what an "average treatment effect" is. In fact, GEE and ML agree on point estimates for independent data. Only with a repeated measures design is there a difference between individual-level and population-averaged effects. You need some critical imbalance in the design to get those different estimates. In that case, GEE does not actually estimate the desired quantity.
The main strength of GEE from a basic modeling perspective is that:


*

*The probability model for the response is a "working model"

*The data do not have to be independent or identically distributed to obtain valid inference.

*The mean model does not have to be correctly specified.

