I'm not very familiar with propensity score matching or causal inference from observational data so I'll focus on answering your question about the use of Cox regression in randomized controlled trials (RCTs).

Randomization has the advantage that it allows to balance the observable plus the unobservable characteristics equally across the treatment groups.

Contrary to popular belief, we do not randomize to balance characteristics between treatment groups. It's false to say that randomization will create equal balance between groups, as this would only occur in the limit (as $N$ approaches infinity). There will almost always be some imbalance between treatment groups in an RCT.

Instead, we randomize to try and evenly distribute future outcomes between treatment groups. Note that I said try - the more variable the outcome, the larger sample size needed to claim with some certainty that outcomes will be evenly distributed. With a large enough $N$, this allows the treatment groups to be exchangeable and causal inferences to be made (assuming other assumptions of RCTs are met as well). Randomization also helps prevent bias by breaking the causal link between any factors that would influence a patient from receiving one treatment over another.

If the goal of randomization is not to balance covariates, why do we use regression models to analyze RCTs? Although covariate imbalances do not invalidate causal estimates, they can decrease statistical power. Researchers often adjust for strong prognostic factors (predetermined before analysis) to decrease the outcome variance between groups, increasing power and decreases the need for larger sample sizes. Here the treatment hazard ratio is the only estimate of interest, and additional covariates used for adjustment should be included based on prior knowledge, not their p-value in the regression model.

For more information on RCT randomization, see this article by Darren Dahly. Much of my answer is taken from this article.

Furthermore there is additional nuance to covariate adjustment in RCTs. Check out this article that discusses the risks and benefits of covariate adjustment in RCTs for more details.

I'm not very familiar with propensity score matching or causal inference from observational data so I'll focus on answering your question about the use of Cox regression in randomized controlled trials (RCTs).

Randomization has the advantage that it allows to balance the observable plus the unobservable characteristics equally across the treatment groups.

Contrary to popular belief, we do not randomize to balance characteristics between treatment groups. It's false to say that randomization will create equal balance between groups, as this would only occur in the limit (as $N$ approaches infinity). There will almost always be some imbalance between treatment groups in an RCT.

Instead, we randomize to try and evenly distribute future outcomes between treatment groups. Note that I said try – the more variable the outcome, the larger sample size needed to claim with some certainty that outcomes will be evenly distributed. With a large enough $N$, this allows the treatment groups to be exchangeable and causal inferences to be made (assuming other assumptions of RCTs are met as well). Randomization also helps prevent bias by breaking the causal link between any factors that would influence a patient from receiving one treatment over another.

If the goal of randomization is not to balance covariates, why do we use regression models to analyze RCTs? Although covariate imbalances do not invalidate causal estimates, they can decrease statistical power. Researchers often adjust for strong prognostic factors (predetermined before analysis) to decrease the outcome variance between groups, increasing power and decreases the need for larger sample sizes. Here the treatment hazard ratio is the only estimate of interest, and additional covariates used for adjustment should be included based on prior knowledge, not their p-value in the regression model.

For more information on RCT randomization, see this article by Darren Dahly. Much of my answer is taken from this article.

Furthermore there is additional nuance to covariate adjustment in RCTs. Check out this article that discusses the risks and benefits of covariate adjustment in RCTs for more details.