Think of MathML more like SVG. You can write it by hand (and in some cases you should), but in most cases you should use a graphical editor (like inkscape), or a library (like D3).
This is exactly the property that makes it actively worse than TeX notation. It makes equations a second class citizen compared to text because you can’t write them comfortably without external tools. MathML is a failure - because of its ludicrous verbosity. The correct solution may not be TeX notation, but it can’t be this bad a step backwards in usability.
In either case, you can use TeX as your input, and if you do, you have to convert it, client-side or browser-side, into something usable by the browser; it's just that if the browser accepts MathML the rendering is faster and/or more convenient, plus you get other options.
Sorry, my claim was about the two notations: that the TeX one is writable, and the MathML is not.
I’m not claiming that a JavaScript parser and complete renderer for TeX is better than a JavaScript parser that renders via MathML. The second option may indeed be more efficient - but making the browser parse a sensible notation for maths instead of an XML crapfest would be better than either.
I don't see why rendering TeX formulas would be slower than rendering MathML. It should be possible to maintain implementations for both variants with similar performance. MathML's parser should be simpler though, cause it's XML, which already has many efficient parsers.