I presume by "m-site" you mean sites which have both a mobile version and a desktop version. Often this occurs where the desktop site was too complicated or involved to allow the developers to turn it into a mobile site using, as you said, media queries and CSS. It was just easier to create a mobile site though, in the long run, it may cost them time and lost visitors who don't click or notice the mobile version.
The usual workflow now is to start with a mobile version of a site and build on that out to a full desktop version. It's easier to do as you have less space on mobile but, as the screen gets bigger, you can add elements to it. Taking a desktop version with all those elements in place can be more difficult as removing elements can destroy a layout making it unusable or unviewable.
Native apps can work well as they blend in to the environment of the mobile device. However, mobile devices typically have their own programming environment, also, thus requiring, for example, Java programming for Android and Swift programming for iOS. Thus the disadvantage there.
Today, there is a movement toward "Web Applications" where the programming environment is shifting toward cross compatible programming over the internet. Standardized web applications make it possible to write code to that standard and know it will work on all devices. This is relatively new but much of such functionality works and works well with modern browsers.
The general topic title would include "progressive web apps".
In our case, for new sites, we have designs laid out starting with mobile and building on that. Thus our programmers progressively add elements and functionality as the screen gets larger without limiting the user in their ability to use the service or perform a task.
One thing to keep in mind is the restriction placed on mobile devices beyond screen size. That usually means connectivity and upload/download speed along with the processing power of the device. Animation may not run well on a phone so if the animation isn't critical to the site then it may have to be eliminated on mobile. Dropped connections while riding in a car or train need to be taken into consideration, too.