First, it seems that you believe that there is only one way of doing things, that you have to have MVC.
You can do whatever you want any way you want. Of course, some things are more appropriate than others for certain scenarios, and some options can do too much work, offer less than expected or demand what can not be offered.
You can make a game, an application of individual use that does not need to contact with server. But I would do these native applications (desktop / mobile), there are those who prefer web for everything. The user will have a grim experience most of the time for no major gain.
You can have a client based on Electron to dispense with the browser, but I do not like the results in most cases.
It's even possible to make a network of communication between several clients, but I do not think it will make up for it.
You have several reasons for having a server and not trusting the client.
If you really need a server, and enterprise applications often need one, then the only popular option is to use Node.js. There are other implementations, but none that I can talk about, I doubt some is useful except in a specific niche.
Of course you will need a database server as well. Possibly other things, including an HTTP server.
If Node is the best option for the problem, you can use it. If the reason for choosing it is because you only know JavaScript, then the reason for the choice is wrong.
From what I understand, most of the time, if the Node caters well, MVC is cannon to kill bird. But it may be changing a bit.
There may be, but I've never seen a good corporate system written with web technologies, at most using HTTP for communication as fallback when you have no other choice. In fact this holds for any application running in the browser. Some applications are saved using Electron or a stream from it. But you can do better native.
Of course not everyone demands something good, otherwise there would not be so many people using something like that. One tip, I've made systems since the 1980s and back then the users were much more productive.
But it is possible, and there are a lot of people doing it.