What statistical methodologies are currently used in clinical trials to estimate treatment effects? For example, are more novel methodologies for obtaining unbiased treatment effect estimates, like Targeted Maximum Likelihood Estimation, being utilized in randomized controlled experiments in clinical trials for new drugs or vaccines?
Or due to small sample sizes and government regulations (depending on the country in question) are statisticians limited to more straightforward and explainable analysis methodologies (regression, exact matching) after designing the experiments?

If you design your clinical trial well (randomly assigning people either a treatment or a control condition) then there probably isn't any good reason to use any fancy statistical methods for obtaining unbiased treatment effects. The whole reason randomized assignment is the "gold standard" is that (if done correctly) it eliminates ANY possibility of bias due to confounding factors. So all you need to do is use a statistical test (like a t test or chi square test) account for statistical uncertainty associated with the fact that you only ran your trial on a sample of people, rather than the whole population.
Methods like regression analysis were developed for situations in which randomization isn't possible or doesn't work correctly. They are often considered a "second best option" to use when you can't use an ideal design.  In reality, of course, randomization can fail, and you might be interested in differential treatment effects for certain groups, or other more complex questions. So there might be specific reasons to use modeling in those cases. But in a plane vanilla clinical trial where the randomization worked and you are only interested in the overall treatment effect I'm struggling to think of a good reason why you would want to go beyond a simple t test.
