Depends on what you want. If you want to achieve the same results that are possible with Unity, you would need to use some more complex opensource engines like Turbulenz and would be restricted to WebGL for now. Rolling your own engine doesn't make sense other than learning and maybe for simple 2D stuff.

If you want to create games in a very efficient way and have a quick time to market supporting a lot of different platforms, there is hardly a way around Unity now.

You're a dying breed my friend. Most software developers, game developers including, have no appreciation for the technical aspects of their field, and no appreciation for the importance of thoroughly learning and understanding the foundations on which their patchy work is standing - they're glorified product managers and all they care about is the end product they can make a quick buck off of until it's superseded by another, not how well-crafted it is.

Using unity or a similar game engine has absolutely nothing to do with how "well crafted" your game is, nor does it tell if the developer wants to make a "quick buck".

I'm all for learning how stuff works and for taking pride in doing things well (showing appreciation for the craft, as you say), but how many people do you see making their own web framework to develop a web app? Maybe it makes sense in some cases, which is true for games as well. And it is even more true if your purpose is to learn, or to make something in a different way from established frameworks.

We stand on the shoulders of giants in lots and lots of ways, and we should make the most of it to continue to build great stuff (games, or anything else).

With that said, I stand by the opinion that knowing how the low-level stuff works makes you, if nothing else, more capable of using the higher-level stuff. We just need to choose the best way of doing what we want to do, with the time, money and tools we have :)

Going from OpenGL to a game engine with a modern lighting and rendering pipeline, physics, an editor, etc. is a huge amount of work, though. It's worth knowing how it all works and being able to write parts from scratch, especially since what's in a game engine is nowhere near satisfying every use case, but there is also no shame in using a preexisting game engine; if programming and (in the case of a one-man show) designing assets for a game is going to take a lot of time as it is, getting bogged down in game engines would be foolish.

Yes, this is why Unity is so popular.

Unity is not a "file format". It's a tool and development environment that reads and writes and edits lots of different "file formats", for which programmers can make runtime and editor components, that artist and level designers who don't know how to program can plug together and configure. And it can create applications for a wide range of different platforms, not just asm.js/WebGL.

It's nice that you're happy coding things up in JavaScript and WebGL just for the web browser, but Unity has a much wider scope than that.

Eventually somebody will write something like Unity that runs entirely in the web browser, and figure out how to make it target all the different platforms that Unity does (once Apple has their brain tumor removed and supports WebGL in their iOS browser for use in anything other than advertisements). But right now there is nothing like it for web development, and it brings a lot to the table that many people really need.