Agile is a many splendoured thing. Which is a good thing. Like any process it can become a straightjacket, and become something that is used inappropriately or inflexibly. Managers that drink the Agile kool aid and don’t understand the underpinning ideas could be a problem, but this is no different to almost any process.
Contrasting it directly with waterfall and asking if it has brought about better outcomes in a quantifiable sense is not an easy thing to do. You need to choose the process that suits the task. There is nothing inherently wrong with waterfall - when used in a project or task for which waterfall is appropriate. In that sort of project Agile would probably be a net drag on productivity.
Agile seeks to address some of the big problems seen in projects that can cause them to go off the rails. But projects differ dramatically in their constraints. However, back when I taught software engineering - in the pre-agile days - I used to warn the students that IMHO the two places where they would fail more often than anywhere else were requirements and integration. I haven’t changed my mind. And Agile is a good way of handling these two. Waterfall fails often because it tries to divine the requirements before anything else. If you have a very well understood problem this can be quite reasonable. But so much software is ill defined. Stakeholders don’t have a full grasp of what they need (as opposed to want) and they can have changing demands. Short cycles with stakeholder involvement helps avoid this. It minimally avoids cries of “this isn’t what I wanted” at the end of the project. Short cycles with some amount of working and tested code helps avoid integration failures, and integration issues can kill a project dead well into its life.
People still underestimate how hard these things are. In addition to the above, Agile style cycles help identify components that are getting out of control early. Waterfall can easily be derailed by the one long pole that was, on paper, progressing fine, but in reality is well behind, or worse, wrong, and incapable of working when integrated into the whole.
All the buzz phrases and things - scrum, stand ups, are just sensible adjuncts to the core problem. You need to always know the answer to one question - are we doing OK? Waterfall makes answering that question very difficult unless you have a comprehensive understanding of every aspect of what you are doing. Which can only come from having done essentially exactly the same thing many times before. Most software projects are not of that ilk.
But back to the OP’s question. Quantifiable metrics? Hard, and IMHO difficult to justify. Unless you run controlled projects with people of much the same skill set and measure the outcomes, you are not going to get proper answers. Companies that have explicitly migrated their practices may report useful thing such as amount projects go over time, over budget or fail to meet the stakeholder expectations. But even these need to be taken carefully.
OTOH, even before Agile became a thing a lot of the ideas were already in use. Just not codified, and often the result of experience by the lead engineers. So, again, direct comparison with pure waterfall may be very hard, simply because the classical waterfall process has been waning for a long time. Some sort of spiral model has perhaps been more common, and you could argue that Agile is (to some extent) a modified spiral with many turns.