You need to distinguish between uses of propensity scores for matching of cases versus for more general adjustments.

The discussion on this page suggests that there isn't much of a use case for propensity score matching. Among other problems, there is seldom much to be gained by throwing away information. Yet that is what matching cases does, with additional problems introduced by using propensity scores for the matching.

That said, restricting yourself to regression to control for covariates can fail if the regression model for outcome, including the treatment effect of interest and the covariates, is incomplete or incorrect. And there's no a priori way to know whether that's the case.

Inverse propensity score weighting provides another way to achieve effective covariate balance between treated and control groups. Cases with a lower probability of getting the treatment get higher weight, providing a more graded balance between treatment groups. That helps to estimate what would have happened had the individuals with the same characteristics been equally represented in control and treatment groups.

You can combine both types of control, via regression and propensity scores, to get what's sometimes called "doubly robust" estimation. If either the regression or the propensity-score model is OK, you can get a reliable measure of treatment effect.

The issues you raise are much more than a couple of paragraphs can cover. Read the Causal Inference book by Hernán and Robins for a thorough recent treatment.

You need to distinguish between uses of propensity scores for matching of cases versus for more general adjustments.

The discussion on this page suggests that there isn't much of a use case for propensity score matching. Among other problems, there is seldom much to be gained by throwing away information. Yet that is what matching cases does, with additional problems introduced by using propensity scores for the matching.

That said, restricting yourself to regression to control for covariates can fail if the regression model for outcome, including the treatment effect of interest and the covariates, is incomplete or incorrect. And there's no a priori way to know whether that's the case.

Inverse propensity score weighting provides another way to achieve effective covariate balance between treated and control groups. Cases with a lower probability of getting the treatment get higher weight, providing a more graded balance between treatment groups. That helps to estimate what would have happened had the individuals with the same characteristics been equally represented in control and treatment groups.

You can combine both types of control, via regression and propensity scores, to get what's sometimes called "doubly robust" estimation. If either the regression or the propensity-score model is OK, you can get a reliable measure of treatment effect--provided, as Björn rightly notes in a comment, that there isn't heterogeneity of unobserved covariates affecting outcome between treatment groups.

The issues you raise are much more than a couple of paragraphs can cover. Read the Causal Inference book by Hernán and Robins for a thorough recent treatment.