Comparing outcomes of two treatment groups: t-test/Mann-Whitney U versus regression I'm a student and had a question about statistical analysis.
I'd like to compare post-treatment outcomes between two groups: group A who received a traditional drug (n = 200), and group B who received a newer drug (n = 100). Some post-treatment outcomes are continuous, whereas others are binary (e.g. % developing organ failure, mortality). The question is: which drug is superior? Group allocation was non-random, i.e. based on doctors' choices/clinical factors - more of an observational study than an RCT.
It has been suggested to me that I should code the drug such that Group A is '0' and group B is '1', and run regression models to evaluate the explanatory power of drug choice on the various outcomes (linear for continuous dependent variables, logistic for binary dependent variables).
My understanding so far is that there are 2 advantages of regression: controlling for confounders, and seeing associations of other variables.
However, I also read that there are 5 assumptions for linear regression models. When I began testing my models for these, I found that my models don't have normal residuals (and so homoscedasticity also becomes hard to test). I don't have the time nor skill to transform / manipulate my data. The other issue is that for some outcomes, there are far fewer data points i.e. n = ~40.
My question is: would a t-test/Mann-Whitney U (depending on normality), comparing mean values for the continuous outcomes between Group A and Group B be okay in this scenario, instead of regression? I've done demographic comparisons and only age differs between the groups; can I report the t-test/MWU P-values with the caveat that any significant results could also be due to differences in age? And for binary outcomes like % organ failure, I could just use 2-tailed tests of proportion?
Would be grateful for some help! I think some versions of this question have been asked before, but I couldn't find a conclusive answer.

The (equal variance) t-test is equivalent to OLS linear regression on a binary predictor (using something called a “Wald test” of the coefficient), and the Wilcoxon test is equivalent to a proportional odds ordinal logistic regression on a binary predictor (using something called a “Score test” of the coefficient). You do not need to separate your thinking into hypothesis testing approaches and regression approaches, since there is an equivalence between the two.
Hypothesis testing and regression often get taught as separate areas of statistics, but you end up unlocking a lot of power when you start seeing standard tests as special cases of regression, as that allows for a lot of accounting for additional variables (such as going from ANOVA with just the groups to ANCOVA that considers the groups AND some additional predictor that you believe influences the outcome).
