Your question rightly acknowledges that throwing away cases can lose useful information and power. It doesn't, however, acknowledge the danger in using regression as the alternative: what if your regression model is incorrect?
Are you sure that the log-odds of outcome are linearly related to treatment and to the covariate values as they are entered into your logistic regression model? Might some continuous predictors like age need to modeled with logs/polynomials/splines instead of just with linear terms? Might the effects of treatment depend on some of those covariate values? Even if you account for that last possibility with treatment-covariate interaction terms, how do you know that you accounted for it properly with the linear interaction terms you included?
A perfectly matched set of treatment and control cases would get around those potential problems with regression.* That leads to the next practical problem: exact matching is seldom possible, so you have to use some approximation. There are several approaches to inexact matching; see this page for some discussion. Matching based on propensity scores, the probability of being in a treatment group give the covariate values for a case, is one frequently used method.
You can also combine matching with regression. You could include covariates in a regression model of matched cases; some argue that you should do this in any event, as noted on this page. You can go even further to potentially include all cases: weighting cases according to their treatment/control propensity scores (inversely) in your regression model. This page nicely outlines matching versus weighting; this page goes into more details.
Both regression and matching have strengths and weaknesses. You need not think of them necessarily as alternatives; combining them intelligently can sometimes work better than either alone.
*Even a data set perfectly matched on the known covariates can't rule out the problem posed by unknown covariates that might affect outcome directly or change the effect of treatment on outcome. That's why randomized trials, which in principle average out those unknown effects, can be so important.